I imagine any mined minerals would stay in orbit and people would “own” that material while never taking possession of it. It’s not too different from what we do today. If you buy gold for your IRA you’re basically purchasing a share of some gold hoard you’re never going to see.
Sending gold down to earth is very cheap. As a bonus, you can also take out a city you are tired of.
In addition to owning orbiting gold, I imagine the real use will be to feed orbital factories, to produce satellites/space stations/rockets in orbit. Everyone on a space station can have a golden toilet.
(edited: typo fix)
I recommend Kerbal Space Program game, it's really fun to do orbital maneuvers there.
Oh, and instead of burning you can use solar sails too. Or shoot lasers or explosives at them to vaporize and give them momentum.
The biggest problem is actually getting to the rocks, chunking them and then not causing a rerun of the movie Armageddon. (Which made changing asteroid direction seem way too easy.) Once the rock is on a trajectory, it will be rather hard to deflect in time, and even then could collide later on. Or hit a newly deployed space station.
Once you fire it, you will end up ruining someone's day, somewhere and some time. It's a weapon of mass destruction.