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.
Just to add to StephenG's answer, and because I think a key point wasn't covered, I'll give this one a go. Warning - answer is probably too long.
1) Yes, it's complicated and the math gets a little intense.
2), As StevenG points out it's a 3-body problem and The 3-body problem has no solution but it can be run on a supercomputer in many cases for millions or tens of millions of orbits, and millions or tens of millions of years into the future with pretty good accuracy. The margin of error doubles with time, so there are limits to such calculations, but within those limits, models of the future of 3-body orbits can be made.
3) and my main point. There's a difference between a stable 3 body orbit (Moon orbits the Earth, Earth orbits the sun), and the unstable model that's typically referred in a 3 body problem like the image below from the n-body problem in Wikipedia).
Because the Moon is comfortably inside the true region of stability inside Earth's Hill Sphere, it's orbit can be considered stable. Ultimately all orbits are unstable, so, when I say it's stable, I mean, stable for hundreds of millions or billions of years. That's a nice round number. Phobos, for example, is expected to crash into Mars or break apart inside Mars' roche limit in about 10 million years. Some people call that an unstable orbit, but 10 million years is a long time. There's no precise dividing line between stable and unstable orbits, it's a term of convenience, more than accuracy.
Using Kepler's laws and the Earth-Moon-Sun as an example, The Earth-Moon barycenter orbits the sun in basically a Kepler orbit. Pretty close anyway. If you want to be a stickler, it orbits the 3 body barycenter, but with the sun's mass ratio to the Earth/Moon being about 330,000 to 1, it's pretty close to just orbiting the sun. Kepler's laws are pretty close.
The Moon does not orbit the Earth in a neat Kepler orbit because solar tides are significant, making the Moon's elliptical orbit pretty stretched and wobbled. The same sun-tides that affect oceans on earth affect the Lunar orbit around the Earth.
This diagrams is exaggerated, so take with a grain of salt, but this is the gist of what the sun does to the Moon's orbit. It's still ellipse-ish, but visibly stretched.
The Moon's apsidal precession completes a full 360 cycle every 8.85 years or roughly 109 synodic orbits, a bit over 3 degrees off where it should be by kepler's laws per orbit. That's over 1000 times the variation of Mercury's precession, which confused astronomers for decades.
This significant lunar precession is almost entirely due to solar tides and it's an example of how solar tides affect the Moon's orbit around the Earth. If Venus or Mercury had moons, those moons would probably have even bigger precessions.
Now, long term changes are far slower as the precession tends to balance out over time. Even with the Moon's significant precession it's still considered a stable orbit and it's comfortably inside the true region of stability inside Earth's Hill Sphere. There's also very little chaotic unpredictability in the Moon's orbit that is typical of a standard 3-body problem. That's not to say there's none, there's still some, but there's also a long term constancy to the Moon's orbit. See here, for a very nice explanation of the 3-body problem.
So, long story short, what happens with a structured orbit (Sun-Planet-Moon), where all orbits are inside true regions of stability, becomes somewhat predictable. Basically the sun steals or saps energy from the planet-moon orbit as an effect of the tidal force. How this happens is tricky to explain and maybe there's an expert who can explain it better or give cool graphics. I don't have those skills.
Now "stealing" is perhaps a bad word. The planet orbits the star and the moon, orbits the planet but that orbit is stretched and squeezed by the stars tidal force. If you think of the Moon as a "bulge" on the planet the solar gravity tugs on that bulge, slowing it down. But the effect is quite small as what it does is both speed it up over 1/2 the orbit and slow it down over the 2nd half, but the de-circularizaiton of the orbit when added up, works out to a decrease in orbital energy over time.
The sun is doing this to the Earth-Moon system now, but it's measurably smaller than the Earth-Moon dynamic that pushes the Moon away from the Earth about 3.8 cm per year. Over time, as the Earth loses it's oceans, and the Moon moves slightly further out, the solar effect will become the primary force, slowing drawing the Moon back towards the Earth, But that's probably billions of years away.
An alternate way to think about this, instead of Sun-Planet-Moon is Planet-Moon-sub satellite or, planet-moon-sub moon. This model we have actual observed data for.
Moons as a rule, don't have their own moons and the reason for this is the same - tidal forces. And on average, the tidal force a moon receives from it's planet are significantly greater than the force a planet gets from it's star, but while the tides are many times stronger, the principal is still the same.
See "Can moons have moons". Dr. Fraser Cain explains what happens, he doesn't get into the specifics of why (math drives away readers). But he explains what happens. From his article
No satellite we’ve sent to the Moon has ever orbited for longer than a few years before crashing down into the lunar surface. In theory, you could probably get a satellite to last a few hundred years around the Moon.
But why? How come we can’t make moons for our moon to have a moon of it’s own for all time? It all comes down to gravity and tidal forces. Every object in the Universe is surrounded by an invisible sphere of gravity. Anything within this volume, which astronomers call the “Hill Sphere”, will tend to orbit the object.
So, if you had the Moon out in the middle of space, without any interactions, it could easily have multiple moons orbiting around it. But you get problems when you have these overlapping spheres of influence. The strength of gravity from the Earth tangles with the force of gravity from the Moon.
Although a spacecraft can orbit the Moon for a while, it’s just not stable. The tidal forces will cause the spacecraft’s orbit to decay until it crashes. But further out in the Solar System, there are tiny asteroids with even tinier moons. This is possible because they’re so far away from the Sun. Bring these asteroids closer to the Sun, and someone’s losing a moon.
The object with the largest Hill Sphere in the Solar System is Neptune. Because it’s so far away from the Sun, and it’s so massive, it can truly influence its environment. You could imagine a massive moon distantly orbiting Neptune, and around that moon, there could be a moon of its own. But this doesn’t appear to be the case.
NASA is considering a mission to capture an asteroid and put it into orbit around the Moon. This would be safer than having it orbit the Earth, but still keep it close enough to extract resources. But without any kind of orbital boost, those tidal forces will eventually crash it onto the Moon.
Earth's tidal forces will cause an object in Lunar orbit to crash into the Moon. (the Moon's gravitational lumpiness doesn't help either - for an entirely different reason, both are factors.)
and the Moon's permanent tidal lock on the Earth makes a lunar orbit difficult. In a sense, there's a permanent heavy side facing the earth and a permanent light side of the moon always facing away from the earth. See article here. That imbalance helps create unstable orbits around the Moon.
The more massive a planet, on average, the more gravitationally spherical it becomes (ignoring rotational bulging, but since most orbits are equatorial, bulging isn't an issue). Moons and smaller planets (like Mars) can be more gravitationally lumpy, which makes orbits somewhat less stable.
But to your question, The solar tidal force does draw moons closer to their planets, but outside of Mercury and Venus, the solar tides are small enough that this isn't much of a factor. When we get better looks at exoplanets, and red-dwarf stars with close orbiting planets, we might find that moons are rare on those planets because of the comparatively larger solar tidal forces, but such observations are probably a ways away. I'm not sure even the James Webb Telescope will identify exo-moons.
Other solar effects, like evaporation/transpiration off a planet's surface would make the planet lighter, so in that sense, a sun can cause a planet to lose a moon if the planet is losing mass due to solar wind. (this effect is quite slow - Titan comes to mind as the best example of a solar-system object that's losing mass due to the solar wind, but even for Titan, this loss is very slow. If Titan had a moon, however, the solar wind would cause it's moon to slowly move away (but Saturn's gravity would draw the moon closer to Titan). In your specific question about Pluto, this is probably a bigger effect than tides, so the sun is probably causing Charon and Pluto to slowly move apart due to outgassing, as it's called.
Also, as a final example of solar effects gaseous pressure from an expanding star would cause a lunar orbit to slow down and cause the moon to crash into it's planet, but gaseous pressure like that isn't expected until our sun goes red giant.
But the answer I think you're looking for is that solar tides will sap energy from a planet-moon orbit around that star over time and quite slowly. As I said above. If somebody can explain the math behind that or show with graphics, please feel free.