nt.number theory - Lower bounds on zeta(s+it) for fixed s

Yes, such conditional results are covered in Chapter 14 of the standard reference - the second edition of The Theory of the Riemann Zeta-Function by E. C. Titchmarsh. This edition has end-of-chapter notes by D. R. Heath-Brown bringing it up to date as of 1986.

In particular a lower bound
$$
|zeta(3/4 + it)| gg e^{-csqrt{log(t)}/loglog(t)}
$$
holds with some $c > 0$, conditionally on RH. See page 384 of the cited reference.

Such results are only known unconditionally for a region to the left of the line $sigma = 1$ that
narrows to zero width as $t rightarrow {pm}infty$. Not coincidentally, the best zero-free region known is also of this form. See page 135 of the cited reference for the best result of this kind.