"Can't remember" and "don't remember" are both possible and they mean almost the same, but the nuance is very slightly different. "I don't remember" implies I'm not putting much effort into remembering it, while "I can't remember" implies I have tried to remember it, but I can't.
In this context it really depends on how strongly you are denying doing the thing. For example, if someone just mixed you up with someone else you would probably use "don't"
A: "B helped me with that."
B: "Huh? I don't remember doing that."
A: "Oh, then it must have been C."
But when denying an accusation you'd be more likely to use "can't".
A: "B hit me!"
B: "What? I can't remember doing anything like that!"
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