That and the fact that both films deal with good and evil.
But they don't,
at all.
The Dark Knight does to some extent (comics do in general; superhero universes are built on this concept of "goodies and baddies"). What's "evil" in
The Godfather?
I can definitely see parallels between
The Godfather and
There Will Be Blood (2007), in that both take on the juxtaposition of personal, intimate stories set against all-American backdrops; Daniel Plainview as self-destructive, distrusting misanthrope (the Michael Corleone of this century, so to speak), etc. And both evoke the terms "epic", "family", "power", "Americanness", etc.
But
The Dark Knight? Ledger's performance is a different sort of performance to those found in
The Godfather. To begin with, he's not playing a real, serious character. He's a concept painted in broad brush-strokes. Daniel Day-Lewis (in
TWBB) is much more of Brando and Corleone's ilk. (And that's not a dent to Ledger; it's a different sort of acting completely.)