Something that has worked for me on sticky, painted wood windows and several other things is parafin wax. It doesn't make a mess and doesn't get on your clothes, etc. It's the same stuff used for home canning jelly, etc. You can get it at hardware or grocery stores at the right time of year. I've found that rubbing some on the surface of my table saw allows the wood to slide smoothly without sticking. It's cheap, too.

Another thought: have you made sure all the screws in the hinges are tight? The door may be binding because of loose screws in the hinges.