Base-Breaking Character: Damian Wayne. While initially hated by fans, he's grown more into this role over time, since more fans have come to like him due to his character development over the films and having grown a bit more humble and open to his teammates over time. On the flip side, he still has his detractors who feel that he's an annoying Marty Stu character who repeatedly gets to show up everybody else and gets too much preferential treatment from the writers. The fact that he's taken down people like Deathstroke, Nightwing, Batman, Talon, and even Superman himself at one point has made him the object of derision from those fans. Damian Wayne is not universally hated by any stretch but he's far from being universally loved by fans in the same token.
Memetic Loser: Dick Grayson/Nightwing in the animated New 52 continuity. He's a skilled fighter in his own right and lest we forget, he was trained by Batman himself. However, a large portion of his fights tend to see him lose a lot, whether it's at the hands of Damian Wayne, Talon, a brainwashed Batman, or Deathstroke. It's gotten to the point where fans are complaining that the writers don't have any respect for Nightwing and that it's Dick's destiny to be the whipping boy of the DC animated movies.
This view is highly debatable. Damian is a character with a multi-story character arc, and he gets progressively more likable with each instalment. Damian is supposed to be a jerk when we first meet him (as he was in the comics) , and he is supposed to warm up over time, like he did in Grant Morrison's run. By the end of Justice League vs. Teen Titans Damian has been sufficiently humbled without outright losing who he is.
What Do You Mean, It's Not for Kids?: These movies pull no punches when it comes to violence, with some pushing their PG-13 ratings to the limit. This doesn't stop uninformed parents from thinking that these movies are perfectly acceptable for children. It certainly didn't stop Cartoon Network from airing Superman: Doomsday (which was aired on their action block Toonami, edited for a TV-PG-V rating) and Batman: Gotham Knight (aired during regular daytime hours, and edited for a TV-14 rating).