If I'm understanding the question correctly, it's referring to a universe that (1) has a spatial topology that wraps around, and (2) has cosmological conditions such that a timelike curve can circumnavigate the universe (in the sense of reuniting with a geodesic that has been at rest relative to the CMB). I assume that "looped" doesn't refer to closed timelike curves (CTCs), which are timelike and whose existence violates causality.
In answer to zhermes's question posed in a comment, no, this would not theoretically require curvature. Analogously, a piece of paper wrapped into a cylinder has no intrinsic curvature. However, the actual cosmological conditions of our universe can and probably do have nonzero intrinsic spatial curvature.
The mathematically simplest cosmology that has loops in the sense defined above is one in which the intrinsic curvature vanishes everywhere, the universe is static, and one or more spatial dimensions are topologically wrapped around. This is essentially a cylinder. In a cylindrical universe, there is a globally preferred frame of reference, which is the one in which the Lorentz contraction of the universe's circumference is minimized, i.e., the circumference is maximized. This does not contradict the foundations of GR, which only say that there can be no preferred frame locally. The existence of the preferred frame means that you can have a non-null result from the twin paradox even if both twins move inertially.
In a realistic, closed cosmology, I don't think condition #2 above is satisfied.