What would happen if Wolverine's adamantine claws penetrated him? Will he die?

He will definitely not die. His immortality is too buggy, and he will be reassembled even if he is cut into pieces.

Super steel is not attached to the surface of the bones like paint, but is integrated with the bones. The bones themselves are the fusion of super steel and immortal bones, allowing unlimited recovery without pressure. Apart from the possibility of the Phoenix Force eliminating immortality or using firepower to blast the X-gene into ashes, there is no way to completely eliminate him. At most, it can seal him and extend his resurrection time as much as possible.