  We derive an effective field theory model for magnetic topological insulators
and predict that a magnetic electronic gap persists on the surface for
temperatures above the ordering temperature of the bulk. Our analysis also
applies to interfaces of heterostructures consisting of a ferromagnetic and a
topological insulator. In order to make quantitative predictions for
MnBi$_2$Te$_4$, and for EuS-Bi$_2$Se$_3$ heterostructures, we combine the
effective field theory method with density functional theory and Monte Carlo
simulations. For MnBi$_2$Te$_4$ we predict an upwards N\'eel temperature shift
at the surface up to $15 \%$, while the EuS-Bi$_2$Se$_3$ interface exhibits a
smaller relative shift. The effective theory also predicts induced
Dzyaloshinskii-Moriya interactions and a topological magnetoelectric effect,
both of which feature a finite temperature and chemical potential dependence.
