They’re both phrases, just have different meanings.
Burn bridges is to piss people off so that you can’t go back and ask they for a favor in the future. Lots of people burn bridges when they leave a job, and regret it later when they need a reference.
Burns my britches means to piss you off. “Chaps my a$$” is another version. It really burns my britches when idiots cut me off while talking on their cell phone.
It’s bridges. Britches are an old term for pants. You may not want to burn those–especially when you are wearing them.
It’s the negative of the well-worn expression, “burn one’s bridges.” That expression, in turn, is a somewhat more modern version of what ancient armies, those of the Roman Empire in particular, were said to do when invading hostile territory. If you got there by crossing a river, a commander would sometimes order the burning of the boats by which the troops got there, so that there would be no turning back, no possible retreat. This ensured that the soldiers would fight their damndest so as to stay alive.
“Don’t burn your bridges” means, then, “don’t cut off all means of escape or retreat,” probably most often used in a non-military context, such as personal advice. Lately, when wars have come, there has often been talk of “an exit strategy” or a way out. Any commander who burns his boats is likely to find himself in hot water.
Idioms: burn one’s bridges
Also, burn one’s boats. Commit oneself to an irreversible course. For example, Denouncing one’s boss in a written resignation means one has burned one’s bridges, or Turning down one job before you have another amounts to burning your boats. Both versions of this idiom allude to ancient military tactics, when troops would cross a body of water and then burn the bridge or boats they had used both to prevent retreat and to foil a pursuing enemy. [Late 1800s] Also see cross the rubicon.
“Don’t burn your bridges behind you” is good literal advice for an army. If they have to retreat, they could be trapped, unable to move from a riverbank.
It’s also good advice for the rest of us in a metaphorical sense. Don’t tell off the boss on your last day of work, for example. Someday you might -might – need to work there again, or he may show up to replace your new boss at the next job six months from now. Always leave people on good terms – you never know when you may meet them again.
Don’t burn your bridges before you cross them.