Can a star orbit around multiple planets or a planet with massive moons?

There's theoretical ways to do it but it's so unlikely as to probably not exist. Assuming you want a system where the planets are in stable orbits around each other. The basic difficulty is the 3 body problem or n body problem. More on it here and here.

For example, a massive planet could (in theory) have a single super-massive moon of similar mass to the planet, though that would probably be considered a 2 planet system. It's not possible for one planet to have two super-massive moons because that wouldn't be stable. In general, planets are many times the mass of all their moons combined, similarly stars are many times the mass of all their planets combined and when that stops being true the system is no longer stable. It's very difficult to generate sufficient mass by adding moons to a system, or by adding enough planets to get more massive than the sun, unless the planets crash into each other but past a certain mass, they'd stop being planets and become a kind of star when enough planets combined.

If we look at gravitational systems, like our solar System, something like 99.7% of the mass is in the sun, so the sun dominates and everything orbits the sun. A few of the larger objects have moons, and curiously, but only cause they're in relatively calm orbits far away from other planets, a few asteroids orbit each other, but the system is very structured around the sun with much smaller stable orbital zones around the planets.

Pluto also has a rather curious orbital system of it's own, likely caused by an impact, where Pluto and Charon are comparatively large and 4 tiny moons orbit around them.

Source:

But these kind of structured systems are only possible if you have a significant mass differential. When you have 3 or more bodies of similar mass and similar distance you get a high degree of mathematical chaos and instability. There are creative mathematical tricks to make it work, but none of them are stable or likely.

This is what 3 similar mass bodies look like, and in such a system, with constant changes, the most likely scenario is that one of the bodies eventually gets ejected. (source, N-body problem above)

There are star systems with several stars, but they are either unstable or contain significant differences in mass. The gravitational structure applies for large planets the same as it does for stars, and there's an article about that here.

You can create stability by having 2 objects orbiting each other and a 3rd massive object quite distant. (Alpha Centauri is that kind of set-up though Proxima Centauri is quite a bit smaller, but it's the same system).

You can even create the heirarchy where you have 2 objects orbiting each other and then 2 more, orbiting each other, but distant so the 2 co-orbitals orbit around each other, and if you do this enough times, you can kind of create enough planetary mass where a more massive star could distantly orbit the entire thing, but it gets very structured and very distant. It's not what I'd consider a normal orbit.

You could also cheat and have several planets in a wildly unstable general proximity orbit and have a star some distance away orbiting the chaotic mess in the middle, but it wouldn't be stable for long.