I'm by no means an authority on this subject. At best I'm an enthusiastic amateur, so take my answer with a bucket of salt.
The static on the radio comes from the redshifted heat of the universe expanding. Even if all energy in the universe has evened out, the expansion is still going on, creating the redshifted heat.
However, so much time will have passed that the distances involved will cause this redshift to be even greater. At some point this static will be on a spectrum too low for a radio to pick up.
Also, even if you were able to tune your radio low enough, as the static is now created outside of the observable universe, there's no new noise that can reach you.