Direct3D in high-end PC gaming shouldn't die any time soon because NVidia GPUs won't support Mantle, But if the performance improvements on AMD side are significant (eg: 9x more draw calls), consumers will shift towards AMD products so NVidia will be "forced" to release a Mantle alternative.
Only developers that really want to push the visuals forward are likely to use it though, small indie teams/studios will probably stick with DirectX (/OpenGL) because it's fast enough and easier (?!) to code.
AMD says Mantle will be open. It is yet to be seen what and how much they mean by "open". Plus, Mantle has been developed around GCN architecture, which may not map 1:1 with NVIDIA's.
But I'm crossing my fingers that Mantle will become the defacto standard for all vendors and all os!.
And like Migi0027 said, it's a win win situation, as competitors (i.e. Microsoft) will be forced to push DX12 forward. AMD GL driver team said they're gonna release extensions to expose all the Mantle functionality for GL.
Which means it will probably make into Core GL in about 4 to 60 years.
Plus, emulating the PS4 & XBox One on PC will be a piece of cake! (same hardware, similar OS, low level access to gpu, what else do you need?, ok I'm simplifying, there are some issues, but still...)