There are lots of instances of using 'up' or 'down' after verbs.
Instances: eat up, drink up, meet up, finish up, start up, fill up, clean up, wipe up, tie up, etc.
What do they add as meaning to a verb exactly?
The best way to think about these is as part of the verb itself. Drink up, meaning "finish your drink" has an acceptedly different meaning to drink.
Up is, of course, a preposition and an adverb implying a spatial difference in height from a low point to a high point. Hence, lift up more closely specifies the action of lift. But in the instances you have described up has lost its original adverbial force.
There are two ways we could classify this up. We could either analyse every occurrence of it and try to draw it a secondary meaning of the word up where it has become a sort of intensifying adverb, presumably with the idea of up to the end, but this in part relies upon etymology, and such a pursuit is not part of your question. I suspect that many of your examples had different etymologies too, and that some were formed by analogy to others.
The alternative method, which I would personally choose – since it's a little more conservative and avoids Byzantine classification – is to say that although they're comprised of two words really they're one semantic and syntactic unit.
This is something of a problem in linguistics which I'm beginning to study more closely now, which boils down to: can you talk about a word or words in isolation, or must you discuss them as part of the sentence, paragraph and discourse in which they occur? Is it valid to chop up a sentence down to the word? Remember, we have certain set phrases in English, such as "to ride a bike" where the riding, you will agree, is very different to the riding of "to ride a horse". While a dictionary might index two meanings or more of "ride", you might consider them to be separate phrases. There are better examples and I welcome comments giving them.
This will apply to down as much as to up, I hope.
All right, I'll answer this because @Cerberus appears determined to shame me into it.
The problem is, up and down when used in phrasal verbs have wildly different meanings, and it is difficult to find common ground among them.
Verb + up is often used as an intensifier or signifier of completion. For example, if you screw up you make a significant blunder. And if you eat something up, you devour it completely. Similarly, down is often used to mark completion: nail it down, get it down (as learning a skill), shoot it down (in the metaphorical sense), and so on.
Still, there are too many other cases. Lock up may or may not qualify. And you can be locked up and locked down at the same time. But the up and down don't have to mean the opposite thing, and often don't. To take something up is different from to take something down. The opposite of take something down is to put something up.
You can screw something up, sweeten it up, mark it up, look it up — all different in meaning. You can knock someone up and knock someone down—two unrelated processes—but while you can soften someone up you can't turn around and soften someone down.
There really aren't any handy shortcuts. You just have to learn when to use up and down in each phrasal verb you encounter.
This is really interesting question. I have a teacher in Russian (he was native Russian speaker) , who has a PhD in English philology. He has divided all prepositions in table and explain how they change the meaning of the main verb in Russian. I believe it is the same in English and up shows action in progress, it carry on the meaning of something temporary or the state of finished activity.It depends on the main verb which one of this is and it colours the activity giving more information. Examples:
eat up ->to take into the mouth and swallow for nourishment; chew and swallow (food) (action in progress)
fill up ->to make full; put as much as can be held into (finished activity).
As others note, I think you'd have a difficult time finding a consistent meaning to the words "up" and "down" in these contexts. You're really talking about a collection of idioms that happen to use the same word. I strongly suspect there is no common etymology.
I heard a joke once about a foreigner, call him Alex, in the process of learning English. An American friend, Bob, invited him to go out drinking with him and his buddies. Alex agreed, but then later called to say he had to cancel. "Something came down," he explained. Bob laughed and said, "I understand, no problem. But the phrase is 'something came up'." "No no," Alex insisted. "Something came down. Wife's foot."
To ruin the joke by explaining it: "Something came up" is an idom meaning that an unexpected event has occurred that interferes with prior plans. Someone's "foot came down" is an idiom meaning they took an uncompromising position. "came up" and "came down" in this case are not opposites; they really have nothing to do with each other. Or as in the joke, what came up could be that someone's foot came down.
If you say, "I took Bob up" (usually followed by "on his offer" or "on that"), that means you accepted his offer or suggestion. If you say, "I took Bob down", that means you beat him in a fist fight.
Sometimes it's an intensifier, but other times it completely changes the meaning. "I screwed the cabinet" means I used screws to assemble it or attach it to something. "I screwed down the cabinet" is similar but can only mean I attached it to something, not assembled it. "I screwed up the cabinet" means I made a serious mistake involving the cabinet and now it is ruined or will require major repair. (If you say you screwed a person that's a mildly vulgar term for sex, and unless I'm missing out on a category of fetish, has nothing to do with metal fasteners.)
Maybe you can find a common meaning or etymology to all these sorts of examples. But frankly I think most have to be considered as individual, unrelated idioms.