On a 2000 Avalon, should I replace the water pump when the timing belt is replaced?

Good idea to do them all at once. The serpentine (tensioner) belt and the water pump will need to come off anyway and if they have not been changed for 5 years or more it would be money well spent if you intend to keep the car around and running.

I know this is many years after the question was asked, but the answer is yes! If you have a timing belt, not a chain, you should get the water pump replaced when you change the timing belt. Many cars with timing belts have direct drive, which means trying to turn over an engine when the timing belt is broken will ruin the engine. If your water pump seizes, you will have to do the repair all over again and risk damage to your engine. I know people who have learned this the hard way.