How to remove folded battery from phone without starting a fire?
a bloated Li-ion/Li-Po battery is very dangerous and should be handled with lot of care.Usually people would discard if the battery is in a dangerous position to be handled. I'm going to assume that your phone is expensive and you wouldn't want to discard it.

Since, you have mentioned that after keeping it out for a while it has stopped making smoke you could do the following:
1.Wear full sleeve cloths, thick fire safety gloves(most important) and some kind of face protection.
2.Always use some kind of prying tool/spudger that is used in mobile repair to remove batteries.

Something like in video could occur if the battery has some charge left in it.

Overall, its a real risk trying to repair the phone. Proceed with caution before you attempt to. Also keep it in a safe place where it wouldn't cause fire.

http://ehs.whoi.edu/ehs/occsafety/LithiumBatterySafetyGuideSG10.pdf