He flatly said that he wasn't her father. She asked him point blank and he said no. My theory is that her father (whether he is Berlin or someone else) is the one who started the fire. Red saved her, gave her to Sam and Sam told her the lie that her father died in the fire.
The question then comes to what is the relationship of Red to the father, and why did Red save her in the first place?
As for Berlin, it starts to make some sense that Lizzie might be his daughter, although it doesn't explain why the Stewmaker had a picture of her. Berlin was a criminal who had a daughter--he double-crossed Red or something similar, so maybe Red set the fire--Berlin escaped, but Red got Lizzie and took her to Sam. Berlin didn't know that, so in order to get leverage or draw him out, Red starts sending body parts to Berlin (obviously not Lizzie's). At some point, Berlin thinks Lizzie is now dead and wants revenge on Red (hence the takeover of the bunker just to get him). But then Berlin finds out Lizzie is alive and sends a lackey to "watch" over her instead of going to her himself, likely because of Red's proximity to her. Enter Tom Keen. Just a theory.
