The up particle in the verb tear up doesn't mean 'into pieces'; that's what it means with tear, because of what tear means. Note that, because English orthography, there are two verbs tear, pronounced differently, one transitive, one intransitive, with completely unrelated meanings. And both of them have phrasal verbs with up. The same up.
- I want them to tear up the agreement. /tɛr əp/ 'rip into pieces' (transitive)
- I want them to tear up at the climax. /tɪr əp/ 'start to cry' (intransitive)
This particle up -- there are others, like the up's in look up, throw up, or stand up -- appears with many phrasal verbs in a completive sense. It's one way to extend the sense of a verb. Phrasal verbs with this particle, like burn up, drink up, eat up, use up, read up, write up, and finish up, all refer to some event or activity going to completion, whatever 'completion' is in each particular case.
So if you're ripping something up, it ends up in shreds. But if your eyes are filling up with tears, you feel like crying. Both of these could be spelled as tearing up. Same up, different tear; but no shreds involved with crying.
Something that burns up burns completely; a house that burns down burns to the ground. Consequently the house burned up means the house burned down, a fact which has not escaped the attention of most six-year-old English speakers.
There are literally thousands of English phrasal verbs with up (because there are many thousands of English verbs, and most of them have several phrasal verbs). Many have the completive sense of up.
When you're thinking of a verb, think about its phrasal verbs, to see what it can mean in context. Like the variegated prepositions and complements that verbs take, the particles they form phrasal verbs with are part of the meaning of a verb. Each one is a puzzle piece, which fits some, and not other, contexts.