In a simple RBC setup, I have a tax process described by the following AR(1) process:
t = c + 0.9*t(-1) + u
where c is a constant and u is a normally distributed shock with mean 0 and variance sigma. I am trying to model a change in the tax regime that is unanticipated by agents.

Is it possible to change the mean of the process i.e. change c to c1 at some future date?

Alternatively, can dynare handle a markov switching process for the tax rate?

Depends on what your are trying to achieve. For IRFS, you can have permanent shocks to the mean. This does not work for simulations or estimation. Markov-switching DSGE models are not possible yet.

A permanent shock simply means you define c to be a unit root process

A shock eps_permanent will permanently move c. You can do this for IRFs, but not for simulations, because due to the unit root your system will move arbitrarily far away from the old steady state/approximation point and thus become very inaccurate.