In reply to Hannahmach:
You should try icing your hands after climbing as a short term remedy, or have a friend/partner take the bones either side of the painful joint and gently pulse the joint apart which can bring relief. For a longer term solution you should look to identify the root cause which could lie in how you use your body when climbing. Poor technique , weak core, arms and shoulders and poor endurance may put an over reliance on your grip. Climbing or hold style set at your local wall may also be a factor.
Worse case voltarol 12hour gel is sheer magic