Burn up or burn down?

My take is that "burn up" comes from some sense that the thing is used up (fuel is used and is gone). "Burn down" means the thing has "burned down to the ground" in that all structure and support is gone.

One might say that "all my stuff was burned up in the fire when my house burned down." You'd be less likely to hear "my house burned up," but it is not totally unheard of usage.

There's also a common third option—"burn through"—which evokes the idea of a wave of fire moving across or through something. "I burned through all my money at that casino in about two hours."

Burn Up:

1 if something burns up or is burnt up, it is completely destroyed by fire or heat : The satellite will burn up as it re-enters the Earth’s atmosphere. burn something up Most of the woodland has now been burnt up.

2 burn something up informal to use a lot of something in a careless way : Most household appliances burn up loads of electricity. He just burns up money!

3 be burning up spoken if someone is burning up, they are very hot, usually because they are ill : Feel his forehead – he’s burning up.

4 burn somebody up American English informal to make someone very angry : The way he treats her really burns me up.

5 burn something up to use energy that is stored in your body, by being physically active: As we get older, our body becomes less efficient at burning up calories.

Burn Down:

1 if a building burns down or is burned down, it is destroyed by fire : She was worried that the house might burn down while they were away.

burn something down The old town hall was burnt down in the 1970s.

2 if a fire burns down, the flames become weaker and it produces less heat

Burn up is usually meant to indicate the usage of something:

We're burning up all our fuel.

Burn up can also mean angry:

That comment is really burning me up.

Or traverse something quickly:

Those racers are burning up the track!

Burn down is normally structural, but it can also mean to go through something (as a fire 'goes through' a house):

My house burned down.

Let's burn down these tasks until they're done.

There is a type of chart called a 'burndown chart' which demonstrates the second option.