Well, when talking about the past they both mean the same thing. However, "No one in their right mind would do that." could be used for past, future, and present tense while, "No one in their right mind would have done that." is strictly for past tense. For example, (Past) P1: I just got a tattoo on my face. P2: No one in their right mind would do that! (Future) P1: I am going to stand in the middle of the highway. P2: No one in their right mind would do that! (Present) P1: Sorry, I can't talk to you right now. I'm robbing a bank. P2: No one in their right mind would do that!

No one in their right mind would do that. (Talking about the present) No one in their right mind would have done that. (Talking about the past)