Depends on what you mean.
There will probably ALWAYS be two major political parties, one (relatively) left-wing and one right-wing. That won't change. So the Republicans will always be around in that sense.
But the CURRENT Republican Party is dying off due to demographic change. That doesn't mean the Republican Party won't win in the future, but they'll have to fundamentally change in some way to get enough support to win a national election. Right now they just don't have enough support among young voters, minorities, non-Christians, gays, and women to win nationally. They need more support from at least some of these groups to be relevant nationally...you can't win just off old white guys.
Whether the change will be becoming more socially liberal, throwing xenophobes/racists in the bushes, going in a more libertarian direction, throwing religious fundamentalism in the bushes, etc., I don't know. But they'll keep losing until they make some sort of major change, which they will sooner or later.