While phrasal verbs are often, if not mostly, unpredictable regarding the semantic function of the particle (up, down, on, etc.), there are a few patterns. Namely, in the instance Martha mentioned–”cut up a tree“–the particle up often connotes completeness.
Compare:
tear a sheet of paper
vs.
tear up a sheet of paper.
Other phrasal verbs are used to make this distinction: cut up, chop up, clean up, wash up, etc.
Anyone know of others that generally follow patterns?
What about “shut up”? What on Earth could that mean other than to shut one's mouth completely?
English doesn't have a partitive to speak of. So it uses different phrases to express completion.
• “He ate the bread.” = He ate of the bread.
• “He ate up the bread.” = He ate all (of) the bread.
Also, you mentioned “tear a sheet of paper”. Some use that phrase to mean “tear out/off a sheet of (perforated) paper”. So it can be tricky even when you're saying it correctly. Of course, context is a great dispute resolver.