In general, no, you don't need to worry about the copper oxidizing. Copper doesn't rust through like iron does, it only forms a layer a few microns thick, then it stops. Aluminum and Silver do the same thing. (It's because their oxides aren't water soluble)
Although the copper oxide isn't as good a conductor as pure copper it's too thin on the fins to make a difference. The only place you worry about it is on the contact surface between the CPU and the heatsink. Most good copper heatsinks come with a piece of tape over that area to keep the copper from oxidizing before you mount it. Once mounted the thermal grease protects the copper.
Gold plating a copper heatsink is a bad idea. It will most likely actually reduce the cooling power, not improve it. In the process of gold plating layers of nickel and chromium are laid down first to improve the adhesion of the gold. Both of them have lower thermal conductivity than copper or gold, so the heat gets trapped by them inside the copper.
Silver isn't used for heatsinks for one very simple reason.....It's wicked expensive!
This topic has been discussed here before, a couple of times in fact:
(Try the search feature)