If I am looking for something in a drawer on Shabbat and then find something muktzeh in that drawer, can I close the drawer?
In terms of moving the object, you should be fine as it is 'kilachar yad' (moving in a abnormal way).
In terms of the drawer, if the muktza thing was the only or most important thing in the drawer it becomes a basis for the muktza item and is muktza as well. If there were other things in the drawer of higher importance than the muktza does not dominate and the drawer remains muttar to move. (Note that in determining the status of basis it matters what objects were in the drawer at the onset of Shabbat independent of what is there now.)