Some givens, because "everyone dies" is boring: The moons stay in orbit, and don't hit each other.
First, the "mundane stuff":
Tides: Goes without saying, insane. Not catastrophic, just really unpredictable. Influence on sea life? Probably not that much.
Volcanism and tectonic activity: High. Volcanoes, lots of earthquakes(not big ones, mind, but lots and lots, probably each time you gain and lose a moon.)
Calendar: Almost worthless over long periods. The length of the day will change as new moons are added and subtracted. How much by? Depends on the size of the moon and its orbital distance. Not a lot a lot but perhaps a day or two over a few decades.
Tidal stresses from switching gravity on the moons themselves would probably make them erupt in their own volcanoes.(They'd almost certainly have liquid cores because tides) Rains of sulfuric acid, big rocks, lumps of ice, nearly anything is possible with just mundane moons made of regular moon stuffs.
Now, the exotics. I don't know much about what planes these moons visit, but coming back from the elemental plane of fire and orbiting close to the planet might have an interesting effect on local weather. Not setting-fires hot, but you know. Problematic ecologically. Also there would be nearly no night due to the giant scary red glow in the sky. A Europa-like moon might freeze and remelt every time it came to visit.
The less often a moon visited, the more likely it would be to have a catastrophically huge eruption and shower ejecta onto the world below. In general it's not if you receive orbital gifts but when and how often you get presents from your moon visitors. Most things would be killed on impact but you might get the occasionally durable houseguest, if there's life on the moons some of the time.