Higher blood pressures and heart rates from exercising when unfit can cause small blood vessels to burst in your lungs as well. It's not usually enough to cause an issue but you can taste it in your breath.
Thanks. I don't have time to read it right now, but you're certain that's a very common thing to happen? I'm overweight and have been jogging and running a lot lately. When I run hard, my nose and lungs burn. I like to sprint in intervals until I can't anymore, after jogging, and then walk for a bit. Repeat. I'm wondering if this is dangerous at all.