After I've been climbing for a while, my finger joints are very sore.
Is there anything I can do to relieve that pain?
This might sound a bit daft - but you could be holding on to the rock too tightly. Your other questions indicate you might be pretty new to climbing, and it is common for beginners to make this mistake.
Primarily, you should be climbing with your legs - pushing your weight up. Legs are used to your weight - arms, and fingertips, are not. If you are steady on your legs - especially if you are able to get your leg straight so you are standing through bone and not relying on a bent knee - then you should only need to use your hands for balancing yourself.
Obviously, there are times when you really do need to grip harder, and even pull yourself up rather than push, but climbing is as much a thinking game as it is physical, and planning your moves goes a long way to preventing fatigue and injury.
Treating joint pain - ice packs probably help. I'd also think about taking an anti-inflammatory, like ibuprofen or aspirin (not at the same time).
Oh friend, I have been down this miserable road just like you. I'll tell you what's at the end of it: swelling of the synovial fluid in the finger joints (which if not reduced leads to arthritis), bone spurs in the knuckles, and a year off the rock. Here's the good news, though: I'm happily climbing again and my fingers don't hurt!
The finger joint pain your feeling is most likely due to your hand position on crimpers. Most of us tend to rely on finger bone position rather than strength when holds get small and walls get overhung. We mash the tips of our fingers into the cimp, hyper-extend the first finger knuckle, and hyper-flex the second, locking out finger bones together and creating a really sweet claw-hand that won't release until we decide to make it. Sometimes (guilty), we'll even wrap our thumb over the first finger nail to crank down even harder. So right, this is really bad for your hands.
If you can force yourself, get to the gym and climb, I kid you not, nothing but 5.6s. Look at your hands on every hold and climb with an open hand: every finger knuckle should be bent roughly the same amount and none of them should be bending backwards. This will help you train your muscles to bear the force instead of your joints. It'll also help you retrain your brain, since by now the bad technique is habit. Once you've got the 5.6s, move to 5.7s and so on. Pay particular attention to your hand position as your chest becomes even with your hands: this is when it's very easy to fall back into bad habits.
Lastly, just some sympathy. Training for good hand position really freakin sucks. It also sucks that there seem to be scores of climbers who can climb with claw-hand and never have a problem with it. At any rate, good luck. I'd do what the above poster suggests in the meantime and hit the vitamin I and ice. I had some luck also with glucosamine/chondroitin/MSM supplements while I was retraining.
Holding on to crimpers is what causes pain in my finger-joints. This is from the Metolius website where they give instructions for hang-board training:
I have often heard that using an open-handed technique also trains crimping strength, while the reverse is not true, crimping does not strengthen the open handed gripping strength (used for holding onto slopers.) I would say, avoid anything that resembles crimping when you are training, and safe that for when you are working on a project.
Rest? Sometimes this kind of pain can be a sign of overtraining.
In the question, the poster doesn't say how frequently he climbs, or how long he's been climbing, so its hard to formulate an exact recommendation.
If you're just feeling a tremendous amount of lower-grade soreness, try reducing the numbers of days a week you climb. I have friends that can climb hard 6 days a week, but I set myself up for an injury if I climb (hard) more than 3-4. When I started climbing, I was only able to climb 2 or 3 days a week.
Like other people said, building more open hand strength will help you minimize how often you need to crimp, but that can be a long term change, and regardless of your climbing level, your body still needs a chance to recover from rough workouts.