There are two questions surrounding pictures hanging on a wall on Shabbos. The first is Binyan and Stira. Is hanging a picture considered adding on to the wall and is taking it off considering taking apart the wall? The Mogen Avrohom (315:2) says that rugs and drapes that hang for beauty are not considered binyan and stira. The Chazon Ish explains that this is because they are not firmly attached and they are often taken down and put back up. The Orchos Shabbos (1:8:14:19) says that Rav Elyashiv says it is not Boneh, even if you never take it down, since it is affixed in a manner that makes it easy to take off.With regard to the picture being Muktza, the Orchos Shabbos (2:19:103) says that if you are very careful with it when you take it down making sure that it doesn’t get damaged, then it is surely Muktza Machmas Chesaron Kis. In the event that it is not so precious and you are not so worried about it, then some including Rav Moshe Feinstein (Igros Moshe OC 5:21:13) that it is not Muktza. However the Chazon Ish says that since it is placed on the wall and not moved from that spot, it has a Din of Muktza Gamur and can’t even be moved L’Tzorech Gufa U’Mikoma.
Are you allowed to straighten the picture if it is not hanging straight? The Orchos Shabbos offers a svara to allow this. he says that since you are moving it in its place, and since this is where it should be, it didn’t become muktza for this purpose Erev Shabbos. However he cautions that this is only a conjecture and you cannot do it without asking a Rov who can provide a clear proof on this matter.