If one is mentally drained, should a person still daven? Even if that person cannot concentrate?

Question

Is there any point at all in davening in such a situation?

Answer 1

Related: Better less with kavannah than more without?
Seems like I should add an answer that tries to balance the answers given already. Read them first, but:
None of the sources given say explicitly that you are allowed to pray without kavannah. They do say that given that we don't concentrate so well these days, we no longer tell you not to daven if you cannot achieve a very high level, and we no longer tell you to go back and try again.
The early sources do say that kavannah is extremely essential. You really have to see the wording of the Rambam [ital. mine] (Hilchos Tefillah 4(14)):

כַּוָּנַת הַלֵּב כֵּיצַד. כָּל תְּפִלָּה שֶׁאֵינָהּ בְּכַוָּנָה אֵינָהּ תְּפִלָּה. וְאִם הִתְפַּלֵּל בְּלֹא כַּוָּנָה חוֹזֵר וּמִתְפַּלֵּל בְּכַוָּנָה. מָצָא דַּעְתּוֹ מְשֻׁבֶּשֶׁת וְלִבּוֹ טָרוּד אָסוּר לוֹ לְהִתְפַּלֵּל עַד שֶׁתִּתְיַשֵּׁב דַּעְתּוֹ. לְפִיכָךְ הַבָּא מִן הַדֶּרֶךְ וְהוּא עָיֵף אוֹ מֵצֵר אָסוּר לוֹ לְהִתְפַּלֵּל עַד שֶׁתִּתְיַשֵּׁב דַּעְתּוֹ. אָמְרוּ חֲכָמִים יִשְׁהֶה שְׁלֹשָׁה יָמִים עַד שֶׁיָּנוּחַ וְתִתְקָרֵר דַּעְתּוֹ וְאַחַר כָּךְ יִתְפַּלֵּל
Proper intention: What is implied? Any prayer that is not [recited] with proper intention is not prayer. If one prays without proper intention, he must repeat his prayers with proper intention. One who is in a confused or troubled state may not pray until he composes himself. Therefore, one who comes in from a journey and is tired or irritated is forbidden to pray until he composes himself. Our Sages taught that one should wait three days until he is rested and his mind is settled, and then he may pray.

You hear his words? Prayer without kavannah is not prayer at all; it just doesn't work. And you are not allowed to do it.
Now look again at the sources brought by the other answers; you will see that they are watering down the Rambam's absolutes, acknowledging our limitations.
What about the OP's question? Well, if a person really really cannot pray, that may be the right answer. But it may also be true that the person could still try to fulfill the Rambam's next words (ibid., 4(17)) [again, ital. mine]:

כֵּיצַד הִיא הַכַּוָּנָה. שֶׁיְּפַנֶּה אֶת לִבּוֹ מִכָּל הַמַּחֲשָׁבוֹת וְיִרְאֶה עַצְמוֹ כְּאִלּוּ הוּא עוֹמֵד לִפְנֵי הַשְּׁכִינָה. לְפִיכָךְ צָרִיךְ לֵישֵׁב מְעַט קֹדֶם הַתְּפִלָּה כְּדֵי לְכַוֵּן אֶת לִבּוֹ וְאַחַר כָּךְ יִתְפַּלֵּל בְּנַחַת וּבְתַחֲנוּנִים וְלֹא יַעֲשֶׂה תְּפִלָּתוֹ כְּמִי שֶׁהָיָה נוֹשֵׂא מַשּׂאוֹי וּמַשְׁלִיכוֹ וְהוֹלֵךְ לוֹ. לְפִיכָךְ צָרִיךְ לֵישֵׁב מְעַט אַחַר הַתְּפִלָּה וְאַחַר כָּךְ יִפָּטֵר. חֲסִידִים הָרִאשׁוֹנִים הָיוּ שׁוֹהִין שָׁעָה אַחַת קֹדֶם תְּפִלָּה וְשָׁעָה אַחַת לְאַחַר תְּפִלָּה וּמַאֲרִיכִין בִּתְפִלָּה שָׁעָה
What is meant by [proper] intention? One should clear his mind from all thoughts and envision himself as standing before the Divine Presence. Therefore, one must sit a short while before praying in order to focus his attention and then pray in a pleasant and supplicatory fashion.
One should not pray as one carrying a burden who throws it off and walks away. Therefore, one must sit a short while after praying, and then withdraw. The pious ones of the previous generations would wait an hour before praying and an hour after praying. They would [also] extend their prayers for an hour.

Notice the "therefore"s. The Rambam is saying that the basic idea of kavannah is that you are not doing something else, you are just talking to Hashem. Stop doing other things beforehand, so that you won't keep thinking about them. And, wait to do other things afterwards, so that you won't start thinking about them early. Just pray.
I bet you thought those "pious ones" were learning musar. Well, maybe they were, but the Rambam's simple explanation is that they were making sure that there was a buffer between everything else they were doing, and their prayer.

Now, let's take a look at a more modern posek who discusses the OP's question explicitly. That's the Mishnah Berurah's Biur Halacha on one of the sources brought by Rabbi Kaii, OH 101:1 [ital. mine],:

ואם אינו יכול וכו' - משמע מלשון זה דאפילו לכתחילה מותר לעמוד ולהתפלל כיון שהוא אנוס שאינו יכול ליישב דעתו וכן משמע לישנא דברייתא ולפלא שלא הוזכר זה ברמב"ם שכתב דין זה רק בלשון דיעבד שאם כיון בברכה ראשונה יצא ואולי באינו יכול גם הוא מודה דזה מוטב משלא יתפלל כלל ועכ"פ נראה פשוט דאם יכול להמתין וליישב דעתו מטרדותיו ולא יעבור זמן ק"ש ותפלה עי"ז דצריך ליישב דעתו
"If he cannot concentrate [on everything after the first blessing of Amidah]"... It sounds as though he is nevertheless allowed to go and pray, since he is helpless... It is surprising that the Rambam does not sound that way, as if he is only yotzei b'dieved... But it is obvious that if he can wait and clear his thoughts before the allotted time passes, that he must do so...

והאידנא אין חוזרין וכו' - לכאורה כונתו אם סיים השמ"ע ולא כיון באבות אבל אם עומד אצל אתה גבור ונזכר שלא כיון באבות כיון דמצד הדין לא יצא בזה האיך נאמר לו שיברך עוד ברכות שלא יצא בהם אחרי דחסר לו ברכת אבות וכי מפני שקרוב שלא יכוין נאמר לו שיברך עוד ברכות שבודאי לא יצא בהם כן היה נראה לי אבל מדברי הח"א כלל כ"ד דין ב' משמע דאינו חוזר אפילו עומד אצל אתה גבור ויותר נ"ל עצה אחרת באופן זה שלא יאמר עוד עתה כלל וימתין על הש"ץ שיאמר ברכת אבות ויכוין לצאת וכשיגיע הש"ץ לברכת אתה גבור יתחיל בעצמו דהלא בתפלה קי"ל דאינו מוציא אלא דוקא מי שאינו בקי משא"כ באבות כיון שאינו יכול לברך אותה בעצמו
"Today we do not go back [even if he failed to concentrate on the first blessing]"... I would think that it means: If he has finished the whole Amidah and had no kavannah for Avos, the first blessing. But if he is standing after the first blessing, and realizes that he did not concentrate, since he is not [really] yotzei how could we say to him that he should say the rest of the blessings, where he will not be yotzei on them either, once he is missing the first one?! [Just because we don't send him back] because he may do no better the next time, is no reason to say the rest of the blessings that he will certainly not be yotzei in! That's what I would have thought. However, the Chayei Adam [seems to disagree]...
I do have an alternate suggestion: Wait for the Chazaras haShatz, be yotzei on the first blessing from him, and carry on with the rest yourself...

Not too encouraging. He seems to hold that you are certainly not allowed to start praying if you can't concentrate on the first blessing - even in our days. And if you didn't, you kinda wasted your time ח"ו and he's not sure what's the point of continuing or why you'd be allowed to do it...

See what I've written in the post referenced initially, from Harav Tzvi Berkowitz shlit"a, about a person trying to "blackmail Hashem", ח"ו: Don't complain if I didn't have kavannah - at least I davened! It may be true that you should go ahead and daven if you really really cannot concentrate at all; I'm not sure from the sources. But be erlich. It is absolutely true that you should be trying to muster what concentration you can. You are speaking with Hashem; don't just "daven up" because you are too exhausted to bother. The level mentioned by that Rambam should almost always be attainable: there is no reason to do other things while you are praying.
[This musar all applies to me too, as well as anyone else. It isn't easy.]
And remember the incredible story they tell about R' Shimon Schwab zt"l (source?): He would drink a cup of coffee if he needed it to be able to say Hamapil with proper kavannah!

Answer 2

The answer is yes, you must pray 3 times a day, regardless of your ability to concentrate. There might be exceptions to this, so please consult your LOR for any personal situations where this becomes a question.

There are many sources that say you shouldn't pray if you can't concentrate, and many powerful words have been written by our holy sages about the supreme importance of kavana during prayer, as well as how terrible it is to not put your heart into your prayer.

Nonetheless, the halacha of our generation is our kavana is so low nowadays that we don't presume we can ever concentrate, so putting off tefilot now so we can concentrate later is an invalid strategy and a person is never allowed to decline shacharit, mincha or maariv due to inability to concentrate.

Sources:

Shulchan Aruch OC 98:2

לא יתפלל במקום שיש דבר שמבטל כוונתו ולא בשעה המבטלת כוונתו [טור בשם ר"מ מרוטנבורג הגה"מ פ"ד מהלכו' תפלה] ועכשיו אין אנו נזהרין בכל זה מפני שאין אנו מכוונים כ"כ בתפלה:

One should not pray in a place where there is something that negates one's concentration, and not at a time that negates one's concentration [Tur in the name of R. Meir of Rothenburg, Hagahot Maimoni on Chapter 4 from the Laws of Prayer - see below]. And nowadays we are not careful with all of this since we are not able to concentrate so much during prayer.

Shulchan Aruch OC 101:1

שצריך לכוין בכל הברכות ושיכול להתפלל בכל לשון. ובו ד סעיפים: המתפלל צריך שיכוין בכל הברכות ואם אינו יכול לכוין בכולם לפחות יכוין באבות אם לא כיון באבות אע"פ שכיון בכל השאר יחזור ויתפלל: הגה (והאידנא אין חוזרין בשביל חסרון כוונה שאף בחזרה קרוב הוא שלא יכוין אם כן למה יחזור) (טור):

One who prays [the Amidah] needs to direct one's intention for all the blessings, but if one is not able to direct one's intention during all of them, one should at least concentrate on the blessing of the forefathers [the first blessing of the Amidah]. If one didn't direct one's intention while saying the blessing of the forefathers, even if one directed one's intention for the rest [of the blessings], one needs to return [to the beginning of the Amidah] and pray. Gloss: (Nowadays we do not return [to repeat the prayer when it is] due to a lack of intention, because [there's a fair chance that] even in the repeating [of the Amidah] it is likely that one will not direct one's intention [again]; if so, why should one return?) (Tur)

Here is the Tur:

דא"ר חייא בר אשי אמר רב כל שאין דעתו מיושבת עליו אל יתפלל רבי חנינא ביומא דרתח פירוש ביום שהיה כועס לא הוה מצלי רבי אליעזר אומר הבא מן הדרך אל יתפלל תוך ג' ימים ר' אליעזר בנו של רבי יוסי הגלילי אומר אף המיצר שמואל לא הוה מצלי בביתא דאית ביה שכרא מפני הריח שטורדו רב פפא לא הוה מצלי בביתא דאית ביה הרסנא וכתב הר"ם מרוטנבורק אין אנו נזהרין עתה בכל זה שאין אנו מכוונין כל כך בתפלה

Rabbi Hanina would not pray on a day that he was angry. Rabbi Eliezer said: after returning from a journey, don't pray for three days. Rabbi Eliezer ben Yose the Galilean says: even someone upset shouldn't pray. Shmuel would not pray in a house with beer, because of the distracting smell. Rav Papa would not pray in a house with fish. Rabbi Meir of Rothenburg wrote: nowadays,we don't follow any of these practices, because we don't have so much kavannah in our prayers anyway.

A little piece of advice that comes from the Chassidic angle. There is davening b'kavana, and there is davening b'dveykut. As has been demonstrated, we have a serious crisis of kavana in our generation, but it seems that we still have the power to daven b'dveykut. This is also considered a function of the heart. So what is it? It means, when you daven, be aware that these are not your words, but these are Hashem's words. As such, treat them like you would treat fine china, pronounce them carefully, with awe, and dignity. If you find you aren't able to focus on the meaning and are bothered by that, you should always able to at least achieve this.

Answer 3

"O YHWH, God of my salvation, I cry out day and night before you. Let my prayer come before you; incline your ear to my cry! For my soul is full of troubles, and my life draws near to Sheol. I am counted among those who go down to the pit; I am a man who has no strength, like one set loose among the dead, like the slain that lie in the grave, like those whom you remember no more, for they are cut off from your hand. You have put me in the depths of the pit, in the regions dark and deep. Your wrath lies heavy upon me, and you overwhelm me with all your waves. Selah You have caused my companions to shun me; you have made me a horror to them. I am shut in so that I cannot escape; my eye grows dim through sorrow. Every day I call upon you, O YHWH; I spread out my hands to you. Do you work wonders for the dead? Do the departed rise up to praise you? Selah Is your steadfast love declared in the grave, or your faithfulness in Abaddon? Are your wonders known in the darkness, or your righteousness in the land of forgetfulness? But I, O YHWH, cry to you; in the morning my prayer comes before you. O YHWH, why do you cast my soul away? Why do you hide your face from me? Afflicted and close to death from my youth up, I suffer your terrors; I am helpless. Your wrath has swept over me; your dreadful assaults destroy me. They surround me like a flood all day long; they close in on me together. You have caused my beloved and my friend to shun me; my companions have become darkness."

Psalm 88.

That sounds much worse than mentally drained. But he did not give up praying. "take no rest, and give Him no rest" (Isaiah 62:6-7).

Answer 4

Well, the Rambam said [Hilchot Tefillah 1:1]:

It is a positive command to pray every day, as it states in Exodus 23:25: Va’avadetem et Hashem Elokeichem – You shall serve Hashem your God... This is prayer, as it states in Deuteronomy 11:13: U’le’avdo b’chol levavchem – to serve Him with all your heart… Our sages [Taanit 2a] offer this explanation: "What is [meant by] service of the heart? This is prayer."

I am not aware of exemptions in halacha depending on how you feel.

Shulchan Aruch 98: "One should not pray... at a time at which he cannot concentrate because his mind is unsettled due to a particular distress, anger, or the exertion of a journey. At present, however, we do not take these [restrictions] into account, because [regardless,] people do not concentrate so well during prayer."