We also hear about poison ivy, acne and other skin inflammations "clearing up".You can use just "clear" (without "up") to mean remove objects from the surface of, as in "We cleared the table" = We took the dishes/silverware/floral decorations, etc. away from and off the surface of the table. It's very precise in its meaning.
"clear up" on the other hand does not work here because it's a set phrase meaning, in a literal way, darkness changing into clarity or brightness as in:
"The sky had been dark and gloomy but now it has cleared up."
(Dark clouds are no longer present, they have disappeared, the sun is shining, the sky is blue and clear)
or metaphorical as in:
"I could never understand algebra, but the latest math professor at school finally cleared up my confusion."
(Dark clouds of confusion about algebra are no longer present in my mind, my understanding is clear. )
"After the party, we had to clear up/tidy up."
No, "clear up" would be wrong in this context.