I've seen several comments on this site that say to check with your local rabbi to get a final decision on what you should practically do.
My question is about how local, exactly, my rabbi needs to be. For example, I have heard that there is a teḥum beyond which my rabbi cannot pass on Shabbat, and that if we are separated beyond this distance, the rabbi's decisions are no longer binding. How exactly does this work? What else do I need to know?
"Local" in this context has nothing to do with distance. It is in contrast to being an "express" Rabbi.
For some reason, congregational and yeshiva rabbis like to give long speeches. On and on and on for hours on end, sometimes. They like to express themselves. That's your express rabbi. Believe me, I've had my fill of express rabbis.
You want a local rabbi, instead. You want someone who makes a short sermon, gets to the point quickly, wraps it up, and gets you to the Kiddush before you and the rest of the congregation get overly hungry, and while the chulent hasn't yet turned into spackle and the kugel hasn't yet burned to black carbon.
"Local" also has a nuance that it's specific for your needs and not generalized. Some rabbis feel that they have to give you every opinion and angle when you ask them a somewhat simple halachic question.
I had this last Shabbat. I ask the rabbi, "Can I use a tissue to blow my nose into it on Shabbat?" So he tells me, "Well ... it depends ... Rabbeinu Tam says yes; Rabbeinun Metumtam says maybe only if it's a specific type of tissue and only if you have a bad cold, not a good cold..."
OMG! By the time he finished his answer, I had already sneezed into my sleeve. Come on, rabbi, all I want is a "yes" or "no".
Seriously, if your rabbi can't give you a straight answer on a somewhat simple question, imagine what things will be like if you have to ask him a tough question! So, definitely, you want a local rabbi.
As we know, תורה לא בשמים היא, halacha is not determined by Heavenly beings.
When we say local, we mean it in the sense of the first definition at Merriam-Webster.com:
having a definite spatial form or location
All we mean to say is that your Rabbi should have a defined physical location, as opposed to angels or כחות נבדלים, celestial entities, which do not have any physical location, and they are therefore invalid for issuing halachic rulings.
As you noted, there is a Halacha that one’s Rabbi may not be beyond the Techum. It is for this reason that the Mishnah (Eruvin 3:5) uses the example of a Rabbi coming to town in its discussion of conditional Eruvei Techumin.
Now, we hold that Techumin are d’Rabbanan (Sotah 30b). By this logic, then, if one is unsure if the Rabbi is in the Techum, we can apply Safek d’Rabbanan l’Kula and therefore you can assume that he is.
Now, let’s say that you call up your Rabbi, or Skype him, or send him an email. In these scenarios, it’s necessary that both you and your phone/computer are within the same Techum, and that your Rabbi and his phone/computer are within the same Techum.
Let’s say that both you and your Rabbi are outside of the Techum; in this instance, you must be within each other’s Daled Amos.
As for the source that it’s until the Techum? That would be Shemos 33:7:
> וּמֹשֶׁה֩ יִקַּ֨ח אֶת־הָאֹ֜הֶל וְנָֽטָה־ל֣וֹ ׀ מִח֣וּץ לַֽמַּחֲנֶ֗ה הַרְחֵק֙ מִן־הַֽמַּחֲנֶ֔ה וְקָ֥רָא ל֖וֹ אֹ֣הֶל מוֹעֵ֑ד וְהָיָה֙ כָּל־מְבַקֵּ֣שׁ יְהוָ֔ה יֵצֵא֙ אֶל־אֹ֣הֶל מוֹעֵ֔ד אֲשֶׁ֖ר מִח֥וּץ לַֽמַּחֲנֶֽה׃ >
And Moshe took his tent and pitched it outside the camp, a distance away from the camp, and he called it the Tent If Meeting. Anyone who sook out Hashem would go out to the Tent of Meeting outside the camp.
As Rashi there notes, Moshe’s tent was 2000 Amos outside the camp. Why is this important? Because if it was any further, it would be outside the Techum, and his rulings would be invalid. Note that based on this source, all of these halachos are true even during the week, not just on Shabbos.
I should note, though, that it’s not that his decisions aren’t binding, per se, so much as them not being considered rulings. Therefore, it’s only rulings issued outside the Techum which are invalid, not the ones previously or subsequently made inside the Techum. It’s a deficiency in the ruling, rather than the person issuing them - in Yeshivish, a chisaron in the cheftza, not the gavra.
Note that none of this is applicable when there are no competent Rabbis in your area. In that situation, you may receive a psak from outside your Techum, as the d’Oraisa of learning Torah overrides the d’Rabbanan of learning Purim Torah.
Edit: I neglected to mention the halachos of sending a shliach to ask the question for you. Since we hold shelucho shel adam kimoso, all of these halachos would apply equally to the shliach, rather than the one who sent him.
