Question: May one discard a paper that contains a pasuk into the garbage?

Short Answer: No, this paper must be placed into sheimos, unless it only has two words of the pasuk or another exception applies.

Question: Is there any basis in halachah to discard old s’farim without burial?

Short Answer: Rav Moshe Feinstein zt”l suggested that one may discard into the garbage old s’farim of Torah SheB’al Peh that do not contain Hashem’s name. However, many poskim disagree and hold that even Rav Moshe did not rely on this leniency.