Swinging anything repeatedly will force blood toward your hands and cause some amount of swelling. If your grip is causing any irritation to your finger joints, or you have joint inflammation caused by another condition like arthritis, this could certainly increase the irritation.
I can't give you any specific advice on your golf grip since I'm a terrible golfer, but when playing racquetball or going to the batting cages, I always find it helpful to use the lightest possible grip when I'm not in the process of swinging, and to take regular breaks to stretch my arms above my head and unclench my hands to reduce the pressure. And like most sports, it might help to try lowering the intensity and increasing the frequency. If you haven't played for a few months, resist the urge to go to the driving range and spend an hour hitting the ball as hard as you can (something will hurt). Buy a smaller basket, lighten up on the swing, and come back later in the week to hit some more.
If it always hurts no matter what you do, get some real advice on your grip, talk to a doctor, and see if an anti-inflammatory medication like ibuprofen does anything for you.