Is OpenPGP vulnerable to the "crypto doom principle"?
The answer (to the title question) is no, due to a slight mixup in terminology. What Moxie says is (emphasis added)


  if you have to perform any cryptographic operation before verifying the MAC on a message you’ve received, it will somehow inevitably lead to doom.


Notice that he says MAC and not signature. The PGP group has discussed this too, so you are not alone in your concern.

In other words, mac-then-encrypt is not the same thing as sign-then-encrypt. Sign-then-encrypt is preferred to encrypt-then-sign. For more discussion on sign-then-encrypt see this question and answers.