No
Even in full auto guns it takes more than one hi-cap (50+rounds, 30 is standard capacity) to heat the gun to the point where a cookoff is likely. High intensity rounds like the .223 get the guns very hot, but it takes well over 100rnds fired as fast as possible before there is any risk, other than getting burned it you touch the hot parts!
The .22lr is the least likely to heat to the point of danger.