There are two senses in which Lorentz invariance is preserved in general relativity, local and asymptotic.
In GR, if you look at the neighborhood of a given point, the local space-time looks like the space-time of special relativity. In such a little patch, you can't violate special relativity without violating General Relativity. So there is a local limit in which special relativity holds.
But on larger scales, curvature does violate special relativity. In particular, you can't find a global time which dilates or contracts, because the slices of constant time don't fit together globally to make a flat spacetime.
But on super-large scales, in a mostly empty patch, the gravitational field dies away. If the gravitational field is localized, the asymptotic space is that of special relativity. In this case, you have a global notion of special relativity--- you can boost or translate these solutions arbitrarily, according to the rules of special relativity. The transformations are symmetries of the asymptotic solution.
The result of this symmetry is that special relativity continues to hold at very large scales. This leads to stringent constraints on solutions of General Relativity, for example, the constraint that they should not be able to get something to move faster than light globally (see this question, and the linked question and answer:
Does a Weak Energy Condition Violation Typically Lead to Causality Violation?).
The OPERA results violate Lorentz invariance in a way that is not allowed by standard General Relativity. Gubser has argued that it requires a violation of the Weak energy condition, the condition that gravity always is attractive on light rays, and Motl has argued further that it leads to causality violations, meaning that if you can go faster than light in a gravitational background, you can go back in time (his cogent suggestion for a physical argument appears here: ergosphere treadmills )
This is one aspect of the asymptotic special relativity restrictions still present in General Relativity. There are more, of course, as any asymptotically flat gravitational background obeys all of special relativity. String theory, for example, is a special relativistic theory which reproduces General Relativity, and it can do this because it's special relativity symmetry is acting on asymptotic regions, on far-past and far-future states which define the string S-matrix.