To solve the mystery, I turned to the research performed by the every-man scholars of Wikipedia and discovered that this puzzle has been solved. The answer lies in the verse prior to "that" being mentioned. Specifically, the 4 things enumerated in the song that Meatloaf will not do for love are:
1. forget the way you feel right now
2. forgive myself if we don't go all the way tonight
3. do it better than I do it with you, so long
4. stop dreaming of you every night of my life
Further, states Wikipedia, "At the song's conclusion, the female vocalist predicts two other things that he will do: 'You'll see that it's time to move on' and 'You'll be screwing around'. To both of these, he emphatically responds, 'I won't do that!'"
So there you have it. "That" actually refers to a number of things, which leads me to believe that the statement "I would do anything for love" is, in itself, a lie. Try not to take it too hard.