To thwart this behaviour (and its undesirable consequences - such as chewed up boxes and cat hair on clothes) I have come up with some solutions that I thought I would share.
All my closets have heavy, sliding mirrored doors which I thought by their very weight would be impossible for a smallish cat to manipulate but I was wrong.
A solution is to install tension rods on both sides (one inside on the left to prevent the right door from sliding and one outside on the right to prevent the left door from sliding). The one on the outside is obvious to the cat and may result in him pulling it away so as to access the door again. If that happens it can be placed at the top of the door. I hesitate to do that though until or unless he figures things out because I worry it could fall on him by accident.
When it comes to bi-fold doors, such as those on a pantry or linen closet, I had to think up something else. Little cat paws fit underneath (in my case) and he can simply pull it open, but try as he might, the door will not open when using the following method. Get a sturdy napkin holder (mine is metal) and place it at the top of the door in the center. A piece of paper or plastic should be put underneath so as to prevent marks when sliding the holder back and forth during opening and closing the door. Make sure you measure the width of the door that the holder has to fit on and that of the holder's inside space before buying one. Mine is 1 5/8 inches.
Opening and closing these doors will require bending and stretching but maybe the incidental workout is a good thing. I try to make these maneuvers when Chico is other-wise occupied as I am sure if he sees me removing the tension rods he will figure out how to do it himself!
Now if he finds a way to use the door knobs on my bedroom, bathroom and front doors, I will have to wrack my brain for more solutions ...