  Motivated by recent experiments on EuS/Bi$_2$Se$_3$ heterostructures, we
study the temperature dependent screening effects on the surface of a
three-dimensional topological insulator proximate to a ferromagnetically
ordered system. In general, we find that besides the chemical potential and
temperature, the screening energy scale also depends on the proximity-induced
electronic gap in an essential way. In particular, at zero temperature the
screening energy vanishes if the chemical potential is smaller than the
proximity-induced electronic gap. We show that at finite temperature, $T$,
and/or chemical potential, $\mu$, the Chern-Simons (topological) mass, which is
generated by quantum fluctuations arising from the proximity-effect, can be
calculated analytically in the insulating regime. In this case the topological
mass yields the Hall conductivity associated to edge states. We show that when
the chemical potential is inside the gap the topological mass remains nearly
quantized at finite temperature.
