From the requirement to place Torah items, and especially the names of G-d, in geniza, it is clear that it is prohibited to leave them where they might come to disgrace.
See, e.g., Maimonides Laws of the Fundamentals of Torah Chapter 6:
כתבי הקדש כולן ופירושיהן וביאוריהן אסור לשורפם או לאבדם ביד והמאבדן ביד מכין אותו מכת מרדות. במה דברים אמורים בכתבי הקדש שכתבם ישראל בקדושה אבל אפיקורוס ישראל שכתב ספר תורה שורפין אותו עם האזכרות שבו. מפני שאינו מאמין בקדושת השם ולא כתבו לשמו אלא שהוא מעלה בדעתו שזה כשאר הדברים והואיל ודעתו כן לא נתקדש השם. ומצוה לשורפו כדי שלא להניח שם לאפיקורוסים ולא למעשיהם. אבל עובד כוכבים שכתב את השם גונזין אותו. וכן כתבי הקדש שבלו או שכתבן עובד כוכבים יגנזו
See also, e.g. here:
The earliest responsum I have seen on the subject is printed in the sefer Be’er Sheva, authored by one of the great Torah leaders of the early seventeenth century, Rabbi Yissachar Dov Eilenburg. He was a talmid of the Levush, and his sefer includes a haskamah from the Maharal of Prague! The Be’er Sheva reports that in his day, it was not uncommon for people to burn the worn-out printed editions of sifrei kodesh. Those who burned the sifrei kodesh claimed that this was more respectful than burying them, because burial often resulted in the sifrei kodesh being unearthed and therefore becoming treated disgracefully.
The Be’er Sheva takes strong issue with this approach, noting that it is prohibited to destroy any type of kisvei hakodesh, and that burning them certainly violateshalachah. The claim that burying the sefarim leads to their desecration is unfounded, he states, because the desecration is a result of not burying the genizahcorrectly. As we mentioned above, the Gemara describes burying in earthenware vessels. If, indeed, all genizah were to be buried this way, argues the Be’er Sheva, then the kisvei hakodesh would never be strewn about after their burial. He concludes that worn-out, printed Torah material must be buried in earthenware vessels, just as one is required to bury sifrei Torah this way. This responsum of the Be’er Sheva is subsequently cited authoritatively by the Magen Avraham (154:9).
See also Halachipedia's article on the topic, which claims there may be some room for leniencies:
There is a dispute regarding printing a paper with Divrei Torah which was done for temporary use and without intent of making it Kadosh.
footnote 8: Rav Asher Weiss (quoted by Rabbi Aryeh Lebowitz on http://www.yutorah.org/lectures/lecture.cfm/739819/Rabbi_Aryeh_Lebowitz/Ten_Minute_Halacha_-_Assorted_Sheimos_Issues) is lenient because of the expense.
They however also cite that:
Organizations should not send out advertisements with the name of Hashem written on them because people may unknowingly throw it out. [21]
Many poskim write that one should not write a pasuk on an invitation as most people simply throw these away. [22]
Pesukim should not be placed in newspapers since the people will throw them away. [23]
footnotes 21-23: Iggerot Moshe YD 2:134-135. Ginzei Hakodesh 7:17:footnote 35 quotes Rav Elyashiv that it would be permitted if you do not write the complete name of Hashem.
Iggerot Moshe 2:135, Ginzei Hakodesh 9:5, Halichos Shlomo Tefilla 20:footnote 72. Rav Hershel Schachter in a shiur on yutorah.org (min 1-2) agrees.Rav Moshe Heinemann, however, writes that the pasuk of od yeshama is only a melitza and not written as a pasuk and is therefore permitted.
Rav Elyashiv in Kovetz Teshuvot 1:115. This is based on the Shach 283:4 who writes that the reason Shulchan Aruch 283:4 says one shouldn't write pesukim on a Tallit is because someone may come to throw it away.
