What happens to the momentum of a star as its mass is converted to energy? Stars move relative to many frames of reference.  We can use the CBR for this question, I suppose, but it may not matter.  Their momentum p=mass x velocity.

Stars lose mass as it is converted to energy. Percentage-wise it isn't much, but it is tons per second, so it isn't negligible.
What happens to the momentum of the star as it loses mass? Presumably it is conserved, so does the velocity of the star increase?  Or does the overall system carry off some momentum with the energy, leaving the stellar velocity constant?
Momentum is conserved. But it is the momentum of the system as a whole. The energy released by nuclear fusion escapes as a result of neutrinos and radiation. If these emissions are isotropic (the same in all directions), or at least with no hemispheric asymmetry, then NO net momentum is removed from the star, it is not accelerated and therefore its velocity with respect to any reference frame remains unchanged. It's momentum in that rest frame is then reduced by the percentage of mass that it has lost. The lost momentum exists in the escaping radiation and neutrinos which will appear anisotropic in any frame of reference other than the rest frame of the star.

In practice most radiation loss is fairly symmetric, if not spherically, then at least not in the sense that more radiation does not emerge from one hemisphere. I am not sure about neutrinos, but I see no reason for asymmetric emission and these are a fairly minor component in any case.