Is there any difference in meaning between:
Closing something up
Closing something down
Burning something up
Burning something down
Closing something up entails physically closing off a space or area. This could be a hole, a room, your eyes etc.
Closing something down is more of a business term, meaning to end a company of sorts or even in its archaic form it can mean to pin something in by enclosing it.
Burning something up is a phrase I’ve never heard of and so I couldn’t give you an idea. Lighting something up, yes, but not burning.
Therefore, any form of physical damage from burning woul be refered to as burning something down.
There’s lighting up – like igniting a cigarette so that it may be smoked. But burning up I hear applied to smaller objects: “After I paid off my mortgage, I burned up all the bank papers.” Or, “My favorite suit was burned up in my closet when my house burned down.”
To me you’d ‘close up’ your own shop, but you could ‘close down’ someone else’s shop, or yourself, if you were a shop. I think.
To me burning something up implies that the thing was, at one point, completely covered in flames all the way to the top. So if my house burned up, I would think that it would be completely blackened by fire, but would still be standing.
If my house burned down, it would be burned to the point that it was no longer standing, whether or not the flames actually reached the top of the house.
Just my interpretation
I’ve heard some people refer to having a fever as “burning up.”
“Burning down” suggests that something is in flames, like a house.
Another use of closed down is, “The sherriff confiscated 10 gambling machines and closed down the bar.”
To me, burned up means completely consumed. “We burned up all the firewood we had taken to the campground.”
For something to be burned down, it has to be standing first. You can burn down a house. You can burn down a tree, but not if you cut it down first.
Then of course, there’s cutting down a tree before cutting it up into pieces.
To me, “burning up” has an image of the flames rising, and “burning down” has an image of the burnt remains collapsing. Different emphases on the same thing.
Aren’t phrasal verbs fun? 
If I ran a store, I’d close it up each night. If I went out of business, though, I’d close the place down for good.
There’s also the metaphoric sense of “it burns me up” – “it angers me greatly”. If an arsonist burned my hypothetical business down, I’d be pretty burnt up over the fact.