>Is it possible to spend a day in future virtual reality that would feel like a week, a month of even a year?

No, but that's also not the effect of time dilation. Your subjective perception of time is always the same, it's everything else that seems to slow down when moving in relation to you.

That said, there are indeed a bunch of gamified relativity simulators out there like this one from 2012 http://gamelab.mit.edu/games/a-slower-speed-of-light/

I don't know of any relativity simulators for VR, but I wouldn't be surprised if there was one out there. I suspect the hard part isn't simulating relativity, but doing it in a way that doesn't cause the VR user to throw up.