It does burn brilliantly. We're currently using old hay to start bonfires for burning off hedge trimmings, which we have an enormous pile of.
We also use it to make little three legged shelters for lambs, like the Isle of Man three legged thing. They can courie down in there and it keeps them right out of the wind. They also leap over them during the mad half hour. The hay will rot down and disappear completely over time.
Last year we had a massive bonfire, well three giant bonfires in fact, to get rid of some mouldy hay. That burnt the grass off completely so we had to re-seed those patches.
We have used old hay as a mulch around the potatoes (laying 6" deep leaves of it to block out the light totally), with polythene over it. There are lots of weed seeds in hay, but they don't seem to be a problem treated tis way.
I wanted to try composting it, but like you, there isn't really room to do so much.