Do Jewish magazines need genizah or can they be wrapped and discarded.
They can generally be discarded. Torah sections should be placed into genizah.
For more details please see Sources:
In Torah law, the obligation is only to put documents with the actual name of Hashem into genizah. The Sages added that even divrei Torah require genizah.
Some write that a Torah sheet that was written for one-time use does not require genizah, and the same idea will apply to Torah in newspapers (see Netziv, Meishiv Davar 2:80, and see also Iggros Moshe, Orach Chaim 4:39, concerning a sefer that is no longer used for Torah study; Rav Moshe writes that his chiddush is contingent on the agreement of poskim in Eretz Yisrael, who did not concur with the ruling).
Another reason for leniency is the fact that newspapers are printed and not written; although many poskim are stringent concerning printed material, some argue that modern non-mechanical printing methods provide greater room for leniency (see also Shraga Ha-Meir 5:61:1).
If there is a chance that there are divrei Torah in the newspaper, the newspaper should be thrown away wrapped in a plastic covering, ensuring that they are treated respectfully.
For a lengthier treatment of the subject [Hebrew] please see here.