I have a dresser that I am trying to swap the knobs on. It's a 5 piece drawer, and I can get the screw out of the knob, but the knob doesn't come off. I can (hopefully) gorilla it off, but I want to be sure it's not common to attach the knobs in some way other than a screw.
A screw through the drawer and into the knob is the only way I've ever seen this done. The knob was probably installed before the finish was completely dried and firmly stuck to the wood. Work it back and forth and it should break free. Maybe tap it gently with a mallet.
Based on some good comments, cutting around the knob with a small blade might prevent pulling off some of the finish.
Your drawer has two pieces at the front: a structural piece, similar to the sides, and a decorative front piece with the finish. The knob is likely held on with a screw. If you can't see a screw behind the knob then it was probably attached to the finished front before the finished front itself was attached, so the screw head for the knob is between the two pieces.
Are there a couple of screws inside the drawer holding the finished front on? If so, removing these will let you unscrew the knob. If the finished front is glued on, it will be more difficult.