This error often comes up in the context of the twin paradox when people claim that it can only be resolved in general relativity because of acceleration. This is not the case.
The only sense in which special relativity is an approximation when there are accelerating bodies is that gravitational effects such as generation of gravitational waves are being ignored. But of course there are larger gravitational effects being neglected even when massive bodies are not accelerating and they are small for many applications so this is not strictly relevant. Special relativity gives a completely self-consistent description of the mechanics of accelerating bodies neglecting gravitation, just as newtonian mechanics did.
The difference between general and special relativity is that in the general theory all frames of reference including spinning and accelerating frames are treated on an equal footing. In special relativity accelerating frames are different from inertial frames. Velocities are relative but acceleration is treated as absolute. In general relativity all motion is relative. To accommodate this change general relativity has to use curved space-time. In special relativity space-time is always flat.
In special relativity an accelerating particle has a worldline which is not straight. This is not difficult to handle. The 4-vector acceleration can be defined as the derivative with respect to proper time of the 4-velocity. It is possible to solve the equations of motion for a particle in electric and magnetic fields, for example.
Accelerating reference frames are a different matter. In GR the physical equations take the same form in any co-ordinate system. In SR they do not but it is still possible to use co-ordinate systems corresponding to accelerating or rotating frames of reference just as it is possible to solve ordinary mechanics problems in curvilinear co-ordinate systems. This is done by introducing a metric tensor. The formalism is very similar to that of many general relativity problems but it is still special relativity so long as the space-time is constrained to be flat and Minkowskian. Note that the speed of light is rarely constant in non-inertial frames and this has been known to cause confusion.
An example would be a rotating frame of reference used to deal with a rotating object. The transformation of the metric into the rotating frame would lead to "fictitious" forces such as Coriolis forces and centrifugal forces. It is not very different from ordinary mechanics.
A simple problem is to solve the motion of a body which accelerates constantly. What does this mean? We don't mean that its acceleration as measured by an inertial observer is constant. We mean that it is moving so that the acceleration measured in an inertial frame travelling at the same instantaneous velocity as the object is the same at any moment. If it was a rocket and you were on board you would experience a constant G force. This problem can be solved in a number of ways. One is to use four-vector acceleration along its worldline which must have constant magnitude. Alternatively, the object is passing constantly from one inertial frame to another in such a way that its change of speed in a fixed time interval seen as a Lorentz boost is always the same. From our understanding of adding velocities we can see that the rapidity r of the object must be increasing at a constant rate a with respect to the proper time of the object T. The rapidity is related to velocity v by the equation
v = c tanh(r/c)From this we derive the equation
v = c tanh(aT/c)For other acceleration equations see the relativity FAQ article on the relativistic rocket.