How do stars affect the orbits of moons? I asked a question previous similar to this, but I'm wondering, can a star make a moon move closer to its planet or further away? How?
Short answer : yes.

Long answer : it's complicated.

This is a classic three body problem, and a real example would be the Sun-Earth-Moon system.

Unfortunately there is no general solution of the 3-body Newtonian gravity problem, so we have to rely on special methods for special cases and even these turn out to be quite complex.

The best answer I could find to your specific question applied to the Sun-Earth-Moon is this one on Yahoo.  As you can see the resulting effect is quite a complex set of periodic effects.  It's small but not negligible.

You can intuitively visualize the effect in such a system by noting that typically to a good approximation the planet-moon system will orbit the star at a considerably larger distance than the planet-moon distance.  The planet and moon themselves orbit around their center of mass.  The relative difference in gravitational forces between the position where the moon is furthest away and closest will give you a rough guide to the order of magnitude of the effect.  A back of the envelope calculation for Earth makes it about a 1% order of magnitude for the Sun-Earth-Moon.

Other systems will be different, but there will certainly be an effect.