The definition of "Lie group" typically restricts to a smooth manifold. If we instead define a "Lie group" to be a topological manifold such that multiplication and inversion are continuous, is the manifold necessarily smooth? Is the smooth structure unique if we want a smooth Lie group?
I believe the answer is yes, since a connected Lie group structure is determined by the Lie algebra, but my search attempts failed.
A Lie group is by definition a group internal to the category of smooth manifolds. So it doesn't make sense to ask for non-smooth Lie groups, just like how it doesn't make sense to ask for a ring which does not have an underlying abelian group.
The right question to ask (which I think is the gist of your question) is how to tell when a topological group has a unique smooth structure that makes it into a Lie group. One answer to this is that every locally compact and locally contractible topological group has a unique Lie group structure (Hofmann-Neeb arXiv:math/0609684). In particular, this means that if a topological group is known to be a topological manifold, then it has a unique Lie group structure.
This MO question discusses this and other facts.