In Greg Egan's "The Moat", this is exactly what a group of people do - breed a new race of humans using non-standard base pairs, thereby meaning that they cannot interbreed with anyone, just the "genetically superior" members of their own race: and rendering them immune to all viruses and preventing imperfection creeping in from quick forays into the bushes with the "inferior" milkman.
I doubt we'd find viruses adapted to the new chemistry very fast. Finding a way to metabolise a new chemical (eg E. coli and citric acid) is a long long way from being able to totally replace all the base pairs in an offspring in one go. The way I understand it, a half-formed mutation (eg replacing only two of four bases) would be line-ending, as the offspring could then not infect any cell, normal or "new". It's not impossible, but it would presumably take a long time to make that jump, perhaps by some more roundabout way, like a dual-stack approach. So, IPv6 vs IPv4 in biological terms then.