Transition from 2D to 3D finite element code, what are the inevitable modifications to be implemented?

So so many places you have to rewrite. The whole mesh handling (accessing faces and edges from cells, neighbors from cells, ...). Shape functions. Dealing with the question of how the normal vector of a face when seen from one cell matches that on a neighboring cell. You will also likely encounter that 3d problems are always much larger and that solvers, preconditioners, data structures, parallelization schemes are all much more stressed and may need to be optimized to be efficient. Everything you have to develop also has to be debugged, and debugging is more complicated in 3d because drawing little pictures on a piece of paper is so much more cumbersome in 3d. It's going to be a lot of work to implement this, much more than just doing 2d simulation.

That's why you shouldn't do it yourself. Rely on one of the big finite element packages out there. Not only do they already have 3d meshes and finite elements implemented for you, they also have nearly everything else you may need in the future already implemented for you. Use your 2d experience as a learning opportunity, then move on to using what others have already built over decades of work.