I'm thinking about renting a wire-feed welder to fix my rusty floor pan but I've never welded on a car before. Do I need to disconnect the battery? Should I keep the gas tank full? Any other tips would be great.

Why not just disconnect the battery anyway just to be on the safe side. I don't think a full tank of gas is gonna help if you burn through the fuel tank or a fuel line for that matter, except you'll just have a bigger fire or explosion! Are you welding near the tank or fuel line? If you are, remove them! It's not that hard. SAFETY FIRST

I'm not going to weld near the gas tank, I just remember hearing that a full tank of gas has less vapor in it, since gasoline doesn't ignite really, but the vapor does.

You might want to buy a MIG vs rent. Prices have really gone down. Use one with gas for fewer sparks. Use at least 25% argon mix with CO2 for fewer sparks and splatter yet. Get a good welding book and practice on sheet metal before starting your project. Controling warpage is a little bit of an art.

I already know how to weld, and I know what I want to rent. I was asking about what precautions to take before I weld on my car.

Hope im not to late but you should always disconnect your alternator when mig welding on your car I always disconnect my battery too.Its alot cheeper than replacing your alt.
Dave

Also, There is a "fire blanket" check with the shop where you are going to rent the welder, It is a pretty thick fire resistant blank that you can put over the tank to keep the gas tank cooler

Be careful of the brake lines as well! You are correct about the tank being full, all the way to the top. That way air is not in the tank, to cause combustion. It is better to be full of water, but a pain in the .......