The person in the video said it was about $\pu{40^oF}$$\pu{45^oF}$. At this temperature propane has a vapor pressure of about 8 atm, so that is the pressure that built up inside the Coke bottle. Once this pressure had built up, it quit boiling.
The fact that he could still fairly easily squeeze the bottle suggests that it was probably not very close to bursting from the pressure, and I very much doubt that even a hot day would have caused enough pressure for it to burst. I once saw the aftermath of a similar experiment with a bottle like that in which the lucky-to-be-alive idiot had used liquid nitrogen to blow it up. The bottle had badly deformed and stretched out before bursting.
Bonus Answer: Why it didn't work
Since propane is less dense than, and immiscible with, water (or Coke), it just floated on top and was the first thing to pour out when the bottle was inverted. If he had held his glove-covered hand over the opening and inverted it so that the propane was on top inside the bottle, rapidly expanding in a confined space, it would have at least had a chance of working.