Of course, the two notions are independent one from the other, but often one of them implies the other under some additional hypotheses. For instance:
A topological vector space is finite dimensional if and only if it is locally compact.
A locally compact topological group with no small subgroups is a finite dimensional Lie group. (Gleason-Montgomery-Zippin)
A locally arcwise topological group admitting a continuous injective map into a finite dimensional metric space is a finite dimensional Lie group. (Gleason-Palais)
I would like to collect results of this sort in general, and especially concerning topological groups. Does anyone know of any other significant examples?