There are several issues with a large propane burner used indoors (or natural gas/methane for that matter - propane is derived from NG, in fact).
The issues with a large open flame on a mobile base inside your house are obvious.
The burner would have to be situated at a distance from any combustibles, similar to safely installing a wood stove.
Then there's the gas itself. Unlike natural gas/methane, Propane is heavier than air, so any leaks in the system will cause the propane to pool where you can't smell it. Then it will find an ignition source, and you can fill in the rest.
Next comes combustion. Propane (and natural gas) combustion byproducts are CO2 (carbon dioxide) and water. Innocuous enough, however:
If there is not enough oxygen available for complete combustion, some
unburned fuel will escape and its energy will be lost. Usually this
takes the form of carbon monoxide being formed in place of carbon
Carbon monoxide can kill you before you know it's there. Even the non-poisonous carbon dioxide can be a problem by replacing the breathable air in your brewing room.
Then there's the more mundane issue of water vapor. I boil off almost 2 gallons in an hour. That liquid has to go somewhere. In your house, that somewhere is into your walls, ceiling, woodwork, etc, where it will make a lovely environment for mold. Imagine dumping 2 gallons of water on the floor of your kitchen.
So while you can do it with adequate ventilation and fire-proofing, it's definitely not something I would do. The risks - especially the heaver-than-air bit - far outweigh the benefits, in my opinion.