First, grass doesn't matter. Wheat matters, and you don't need grass to grow wheat. For those that don't know, you need to eat or you'll die and wheat can be used to make bread. But there's plenty of stuff to eat, including renewable zombie flesh.
Second, while there are a few blocks that aren't renewable, there are plenty that are. Wood and cobblestone are completely renewable. And a lot of blocks aren't renewable but never used up. You should have just as much dirt in the world when you end as when you begin, unless it gets blown up by creepers or thrown into lava.
And you hardly need to build on a mountain and then cut away the mountain. Just build up and outward.
Eventually you should end up with people using only wood tools and be completely out of interesting materials. It would just be boring at that point.