Modern climbing gear seems to be pretty resilient to moisture. Of course the lubrication of water affects every point where friction occurs to some degree. So the safe thing to say here is be careful and move slowly 'til you get the feel for it. In my experience, I end up worrying about it much more than it actually affects the operation of my climbing set up. I just don't think the slight lubrication of soaked textiles makes a whole lot of difference at the relatively low forces experienced in most climbing systems. Much more frustrating (in my experience) is how the rain affects my grip on ropes, tree parts, and saws. That coupled with slippery footing on tree parts, are the real concerns with climbing in the rain.
Having said all that, rigging operations are a different story. Running a rigging rope through/around a friction brake of any kind can be drastically altered depending on the rain/moisture. I think the greater forces involved mean that the moisture present can and does reduce friction, sometimes drastically. I remember once it was drizzling out. The kind of rain that doesn't soak you very quickly. But, if your in it for several hours, you're gonna get drenched without realizing it. We were lowering spar wood with a port a wrap. We wrapped the barrel of the device for normal lowering of a piece that size. We were shocked when the ground man was unable to slow it's descent and it sailed down to the ground at full speed - the rope and port a wrap were wet. There were no obstacles or dangers so, no harm no foul. But, it woke us up. For the next piece, the groundie added an extra wrap. Well, the log dead-stopped and shook the shizzle out of the climber - spurs kicked out, tweaked his back. Again, we were shocked. We analyzed the situation and the only thing we can figure out is that when we lowered the first piece, the rope was soaked. It had been lying out in the drizzle for a while. We think during the sailing down of the first log, the water got wrung out of it. The second log followed shortly after. It was only drizzling so the rope was still relatively dry, so the extra wrap was too much.
We chunked the rest down without rigging. Be careful when rigging in the rain.