Not sure what the playoff percentage is for how many games go to OT. I would guess 20 percent.
I can tell you tomorrow up to a few years ago. I have that number somewhere, though it hasn’t been updated for at least three or four years.
I meant to follow up on this and forgot about it. I have a spreadsheet showing NHL playoff overtime stats, but it hasn't been updated in several years. As of 2014, there had been 4,022 NHL playoff games played and 790 of those (19.64%) went to overtime; of those, just under 80% ended in the first overtime, just over 14% ended in the second, 4.43% ended in the third, 1.139% (nine games ever) ended in the fourth, and 0.253% ended in either the fifth or sixth overtime (two games ever have ended in the fifth, two games ever have ended in the sixth).
It would seem notable that the two six-overtime games both took place in the 1930s, when the ice was not resurfaced between periods, but on the other hand both five-overtime games took place since 2000.