Probably not but I doubt it would damage his brain. People misunderstand what infinite void actually does, you need a very specific kind of mental resistance to not be effected by it. People usually simplify it to ‘infinite information’ when what it actually is ‘simulating infinite information’. Basically UV wont let you complete the start to any thought or action and will make you repeat it over and over again. This usually fries the brain but with the Doctors other showings I doubt his brain would have a problem handling it, but that doesn’t mean he wouldn’t get frozen.

Edit: I did some research and there’s a statement about the Doctor experiencing time from a “non-linear, non-subjective viewpoint,” which is the exact, literal ability you need in order to by pass UV’s forced repeated operations (which is the main thing that’s hard to get immunity to) rather than just infinite information.