Is there a phenomenon in physics, whereby a planet could be experiencing accelerated passage of time relative to the rest of the universe?

Spooky action at a distance? Or my favorite YouTube video on dimensional travel Ten Dimensions Explained

What is it you want to speed up? Whatever it is, you plant it here, and return to close in orbit your black hole for a day or so, or as long as necessary. Can a galaxy be so massive that light cannot escape it?

I don’t think so. A gravitational field or acceleration will slow time down but can’t speed it up.
PS Your question implies there is a standard time operating in the universe but there isn’t. It is literally all relative.

It would depend on your POV. If you were on the world where the passage of time was in question, it might seem normal time to you, but from your perspective, time off planet would be slowed. If you were an off world observer of the planet, then yes, time on planet would be accelerated. (Of course, no matter where you were, or what your observation was, the speed of light would be the same.)