Moshe Rozali wrote a very sane text about the importance of Lorentz symmetry for the search for the fundamental laws of Nature:
So Seth Lloyd of the quantum computing fame wants to believe that the world is a quantum computer. Robert Laughlin wants to imagine that quantum gravity is an example of the fractional quantum Hall effect. Other people have their own areas of expertise, too. Peter Woit wants to believe that a theory of everything can be found by mudslinging and defamations while Lee Smolin wants to believe that the same theory can be found by selling caricatures of octopi to the media (following some subtle and not so subtle defamations, too).
Moshe Rozali correctly tells them that if they are going to ignore the Lorentz symmetry, a basic rule underlying special relativity, they are almost guaranteed to fail. Lorentz symmetry is experimentally established and even if it didn't hold quite accurately, it holds so precisely that a good theory must surely explain why it seems to work so extremely well in the real world.
Moreover, the state-of-the-art theories of the world are so constrained - i.e. so predictive - exactly because they are required to satisfy the Lorentz symmetry. Because of this symmetry, quantum field theories only admit a few marginal or relevant deformations. If you assume that they make sense up to extremely high energy scales, you may accurately predict all of their low-energy physics as long as you know a few important parameters. Such a "complete knowledge" of physics in terms of a few parameters would be impossible in non-relativistic theories.
String theory is even more constrained than quantum field theory: it has no adjustable dimensionless non-dynamical parameters whatsoever. In some sense, you may view string theory as a tool to generate privileged quantum field theories with some massless spectrum and infinitely many very special, selected massive fields with completely calculable interactions. So all the Lorentz constraints that apply to quantum field theory can do the analogous job in string theory, too.
However, in string theory, the character of Lorentz symmetry is even more direct. The very short distance physics of string theory is pretty much guaranteed to respect the Lorentz symmetry. Whenever you look at regions that are much smaller than all the curvature radii of a D+1-dimensional spacetime manifold, the dynamics of a closed string reduces to a collection of D+1 free scalars on the worldsheet which manifestly preserves the Lorentz symmetry. And one can show that the interactions respect it, too.
Open strings may violate the Lorentz symmetry spontaneously, for a nonzero B-field or a magnetic field on the brane, and one can enumerate a couple of related ways to spontaneously break the Lorentz symmetry with the presence of branes and their worldvolume fields. But none of these pictures ever hides the fact that the fundamental theory behind all these possibilities is Lorentz-invariant.
There's a lot of confusion in the public about the fate of the Lorentz symmetry in general relativity. Be sure that the Lorentz symmetry is incorporated into the very heart of general relativity.
In fact, even in spacetimes with a lot of curved regions - such as spacetimes with many neutron stars or even black holes - one can use the tools of special relativity in many contexts: either in very small regions that are much smaller than all the curvature radii, or in regions that are much larger than stars and black holes. In the latter description, the stars and black holes may be viewed as local point masses or tiny disturbances that follow the laws of relativistic mechanics at much longer distances, anyway.
So if someone completely neglects Lorentz invariance, the player that became so essential in 1905, he shouldn't be surprised if theoretical physicists simply ignore him or her. It is not necessary for a theory to be Lorentz-invariant from the very begining. But a theory only starts to be interesting as a realistic theory of our world after one proves that Lorentz invariance holds exactly (or almost exactly).
I am personally convinced that theories that try to break Lorentz invariance by small effects are not well-motivated. But even if I insist on the things that have been established only, the "at least almost accurate" Lorentz symmetry that has been demonstrated is an extremely powerful constraint on any theory. If you invent a random theory for which no reason why it should be Lorentz-invariant is known, it is extremely likely that the Lorentz symmetry doesn't work at all and the theory is therefore ruled out.
There are actually approaches to string theory that are not manifestly Lorentz-invariant. For example, the BFSS matrix model, or M(atrix) theory, is a 0+1-dimensional quantum field theory - a U(N) gauge theory with 16 supercharges. You can also say that it is a quantum mechanical model with many degrees of freedom organized into large Hermitean matrices. It resembles non-relativistic quantum mechanics, with some extra indices and a quartic potential.
There is no a priori reason to think that such a seemingly non-relativistic theory - whose symmetry actually includes the Galilean symmetry known from non-relativistic physics - should be Lorentz-invariant. Except that one can defend and "effectively prove" this relativistic symmetry by arguments based on string dualities. Although it can't be completely obvious from the very beginning, the original BFSS matrix model describes a relativistic 11-dimensional spacetime of M-theory. But the relevance of the matrix model for M-theory only began to be studied seriously when arguments were found that these two theories were actually equivalent.
You simply can't expect your non-relativistic model to be equally interesting for physicists if you don't have any evidence that your model respects Lorentz invariance - or if it even seems very likely that it cannot respect it. Physicists would be foolish to treat your theory on par with QED or the BFSS matrix model because it seems excessively likely that your theory can't agree with some of the basic properties of the spacetime we know.