I assume removing a face means here: select a face, remove all its vertices and all their incident edges. (Otherwise I don't understand what happens to edges when one of their endpoints is removed.)
Then the answer is no. Some small counterexamples are Graph 226 and Graph 160 (House of Graph numbering). Pictures (courtesy of HoG) below. Removing the central face from each of them leaves a triangle and a square, respectively, so the vertex connectivity goes down to two in each case.
