It is very much problem dependent, but the fact that it is coupled doesn't matter.
In other words, if you have BC's properly defined for the thermal model, ignoring the mechanical aspects, and if you have BC's properly defined for the mechanical model, ignoring the thermal aspects, you have BC's properly defined for the coupled system. It's not a bad idea to take a "divide-and-conquer" approach to coupled problems, setting up and solving each problem individually, then doing a one-way coupling, then doing the full coupling.
In general, there's a few heuristics to keep in mind when solving such problems:
- For each submodel, in each dimension, you need the same number of BC's as the highest order derivative in the problem. For the heat equation, this is typically 2nd order in space, and thus you need two BC's specified
- What type of BC you use (Dirichlet, Neumann, Robin) and where you apply it depends on what you are modeling -- remember, you can apply BC's at $\infty$
- For the heat equation, and others, you can't have Neumann type BC's on every boundary in a steady state problem without add'l information. You need to specify the temperature somewhere or sometime (Note: this isn't necessary a hard-and-fast rule for all models you will encounter, but if it comes up, take a second look)