I've faced a number of weird openings in my time, some so weird I'm not sure they even qualify as openings. I'm pretty sure they're designed to skip the opening phase altogether, and start the middlegame as soon as possible.
For example I faced 1.e4 e5 2.Nf3 Bc5 this evening, and after taking on e5, I spent the next few moves making sure I didn't have to return the pawn, going so far as to miss a strong tactical shot around move 6. (I did win the game btw).
Of course this isn't as bad as the Irish Gambit (1.e4 e5 2.Nf3 Nc6 3.Nxe5) which I have played against at least twice in serious games, but coughing up a pawn for little compensation makes little sense to me, even as some kind of psychological ploy.
But having said that there are over 250 games with 2. ... Bc5 in my database, so giving up a pawn this way must appeal to some folks.