Well, I once worked on a PC that had mice living in it, and ya did their business on the GPU and it did work after cleaning and sealing off where they were getting into, an open expansion slot in the case, I installed a cover, it was a PC for a auto shop out in the country so it was common to have field mice there and it didn't seem to matter what they did, they even stated to me they spent thousands and they just come back.
I would not power it on, get a cue tips or cotton swabs, or even some paper towel and some 99% alcohol and clean it, try not to dump to much on it if you do not take the cooler off, if you do take the cooler off and the thermal pads, feel free to give it an alcohol bath, submerge it, and a old tooth brush will do wonders, let it dry in front of fan for at least 24 hours, no fan, 48 hours or more.
If you do have to put the PC back where the rats/mice were, definitely seal up all the areas where they can get into it, I know storage units do have mice problems, I've worked on many PC equipment and camera equipment with mice damage around this area anyway and they never seem to do anything about it.
Good luck!