No it won't damage the cards (could put a significant load on the PSU and in due time it could fail if it is being overloaded). You would just notice that you can't overclock the cards and also that the performance isn't as good as you would have expected. I'd say open a new thread in the power supply section and ask if your PSU has enuff power for two 7800GTX's as well as your other stuff.
My guess is that it will be almost pushing it though...My computer pulls probably 450 or 500 watts at max and i only have a 850xt although it and the processor are overclocked
Impossible. I was running a x850xt oced with a p4 @ 3.6ghz on a 300W psu for some time. My estimate for that system is 265W max