  We consider topological superconductors and topological
insulator/superconductor structures in the presence of multiple static vortices
that host Majorana modes and focus on the Majorana tunneling processes between
vortices. It is shown that these tunnelings generally lift the degeneracy of
the many-body ground state in a non-universal way, sensitive to microscopic
details at the smallest length-scales determined by the underlying physical
problem. We also discuss an explicit realization of the Jackiw-Rossi zero-mode
in a topological insulator/superconductor structure with zero chemical
potential. In this case, the exact degeneracy of the many-anyon ground state is
protected by an additional chiral symmetry and can be linked to the rigorous
index theorem. However, the existence of a non-zero chemical potential, as
expected in realistic solid state structures, breaks chiral symmetry and
removes protection, which leads to the degeneracy being lifted. Finally, we
discuss the implications of our results for the collective states of many-anyon
systems. We argue that quantum dynamics of vortices in realistic systems is
generally important and may give rise to effective time-dependent gauge factors
that enter interaction terms between Majorana modes in many-anyon systems.
