No, they do not. 1. "Don't fail me" is a phrase most likely said after a very important request/order has been made. It is a little intense. It is similar to "Do not mess this up" and "Do not do this incorrectly". "Don't fail me" implies that the speaker will be very upset if the other person makes a mistake. Or, at the very least, the speaker will be disappointed in the other person. 2. "Don't make me wrong" sounds incorrect. Do you mean to say, "Don't get me wrong"? If so, "Don't get me wrong" means, "Don't misunderstand". For example: "I hate that dog. Don't get me wrong- I love dogs, but that one is very mean."

What Quinn said is correct. One minor addition. If the second sentence is related to the first, it may mean "don't make me look bad." Basically if you don't do a good job at work (or something else) , it makes your superiors or the team look like they failed as well.