If I wanted to translate the prepositional phrase "on the bed" to Chinese, I know to say "床上", but I don't know what the translation would be for saying "above/over the bed".
From what I can gather, 上 is used for both "on" and "over". Is there another word that removes the ambiguity, or some modifier?
"above/over the bed" can be translated as "在床的上面" or "在床的上方" without any ambiguity.
I would suggest you need to add more context such as:
挂在床上
Hanging over the bed
or
漂在床上
Hovering (floating) over the bed
Even in English above the bed still requires more info to make a complete sentence. I think the context is important here.
There may not be a definitive way to convey these differences.
"在床上" is "on the bed", to really eliminate the ambiguity, you could say "就在床面上" ("right on the surface of the bed") or "就在床单上" ("right on the sheets of the bed").
And use "在床的上方" to say "above the bed", or more specifically, "就在床上方的空气里" ("right in the thin air above the bed").
It's touched by the other answers but not as explicit: in Chinese, disambiguation is mostly done by nouns/verbs rather than prepositions. So the answer is to use distinctive verbs depending on the context: 躺在床上(边/面) vs. 挂在床上(边/面/方) vs. 悬在床上(边/面/方). (躺在床上方 is also correct but against idiom.)
I would use 上 or 以上 for on top or (immediately) above, and 之上 or 上方 for over (more remote).
上面 refers to a surface, e.g. the top of a table or bed.
