Burnt up implies that the structure still stands. Burnt down implies more that whatever was on fire was leveled in it.
For example, you would call a car that had been on fire a burned up car, rather than a burned down car. A strip mall that was collapsed in a blaze would more likely be refereed to as burned down, where as if the structure remained with only the inside walls damaged, it would more likely be called burned up. This is a very useful was to gauge just how severe fire damage is to a structure.
This has been: rural language with Das. I hope you enjoyed today's lesson and tune back in tomorrow, where I discuss the sexual differences between a sheep and a fat girl.