Question: Am I allowed to dispose of divrei Torah sheets such as these?
Answer: The Gemara (Makkos 22a) writes that one who destroys one of the sheimos, names of Hashem, has transgressed the negative commandment of You shall not do this to Hashem your God. (Devarim 12:4)
Rambam takes this a step further and writes (Yesoidey Hatorah 6:8) that one must never burn or destroy Torah writings (except for those written by a heretic, etc.) The Netziv (Meshiv Davar 1:80) limits this prohibition to materials that were written to last. Thus, one would be allowed to dispose of one’s rough notes. Based on this, R’ Yitzchok Yaakov Weiss (Minchas Yitzchak 1:17) permits disposing of newspapers that contain Divrei Torah. R’ Shmuel Wosner (Shevet Halevi 5:162) however, writes that they require burial.
R’ Moshe Feinstein (Igros Moshe OC 4:39) writes that often too much is printed, especially in school settings, which just compounds the problem. He writes that if the gedolim in Eretz Yisrael would agree with him (they didn't..) he would have allowed one to even dispose of a worn Gemara, providing it didn’t contain sheimos. Once a sefer becomes unusable, it loses its kedusha somewhat.There is much debate about placing such papers in recycling. R’ Moshe Sternbuch (Teshuvos Vehanhagos 1:554) maintains that the ideal solution is to put them in a plastic bag before disposing of them. This way, one is treating them respectfully.