Relativity will probably suffer the same fate as classical mechanics: true for small values of v. In relativity's case, the small value will probably be about 0.99999999c.
Given what we know today, it seems that it's much more likely that special relativity will remain more or less as we know it today (a good approximation for relatively flat regions of space-time and non-accelerating bodies), and that GR will be a good approximation for something like large regions of space-time and huge masses (where "large regions" may mean anything larger than a few Planck lengths, and "huge masses" may mean anything greater than a few micrograms).
The Unruh effect can happen in flat spacetime, for example, and is a useful check on Hawking radiation (in a black hole curved spacetime).
It's also interesting in understanding the weak equivalence principle in detail; very loosely, a small-mass object can rest quietly on the surface of a non-compact, non-spinning, spherical mass eternally, while it is at least very difficult to keep the same small-mass object accelerating uniformly in flat spacetime for even fairly short (compared to say the age of the universe, which is hardly eternal) finite times.
