For Lie groups, the only one with this property is $S^1$.
To see this, consider a one parameter subgroup given by exponential of a vector in the lie algebra, and the closure of this subgroup should be abelian.
Since the only abelian lie group with the property you are asking is $S^1$, this concludes.
For other groups, I believe it should also be true, but I don't know. It is interesting to look at the adding machineThe argument above, which gives you an example such shows that except from if the neutral elementgroup is not abelian, then it does not hold (since the closure of the an orbit of each element, is homeomorphic to the group itself (an abelian closed subgroup), but I don't know if abelian topological groups are all known (the ones I am not wrong). know, the only one verifying your property is $S^1$).