The current version runs fine on my AMD card. I don't expect they'll be deviating too greatly from the current rendering codepath, so I wouldn't expect problems.
There's a bizarre "API versions are everything" mentality that's pretty prevalent here. Some people won't even buy a D3D9 game regardless of how well it looks or how well it performs simply because it's not D3D10/11. I'm not sure how these people figure out what OpenGL games to boycott based on API version, with all its real-world conformance and extension complexities.
Which AMD card do you have? When you say the current version runs fine on your card, are you talking about Doom 3 or Rage?
Is the Doom remake just a tweaking of the old version or extensively modified with Rage code? If it is the latter I am more concerned AMD cards will run it poorly, which would be a shame.