It can be used as a replacement for "don't blame me!", but it has slightly broader possibilities. It would be used when either blame, responsibility, or attention, is getting re-directed on to the speaker, and away from existing, or appropriate, focus of the conversation. That focus was someone, or something, other than the speaker.
To better understand how it might be used, one could replace it by saying:
"Don't redirect the focus to me!" or "Don't let the focus get redirected towards me!". These are pretty much a literal rephrasing for meaning, although it would rarely be said like that. However, saying it this way, you can see how "Don't turn this back on me!" could be used.
E.g. Joe says to Mary: "Yes, I know, I was late! But you were supposed to pick me up!" You can see how Joe attempted to re-frame the argument to deflect blame from himself, and on to Mary. Mary might respond "Don't turn this back on me!" as an objection to that redirection of the argument.
The re-framing could be implied, rather than direct. This could be the situation where two (or more) people are involved in a risky undertaking (either legitimate, or illegitimate), and one person is assisting, rather than directing the action. The assisting person may not want the consequences to be directed at them, rather than the active participant, when the active participant might be able to redirect the consequences on to the assister. E.g. Joe and Mary are sitting in a car outside a store. Joe says to Mary: "Give me your gun." And then proceeds to rob the store. Joe gets back in the car. The police arrive before they can leave. Joe gives the gun back to Mary, and tells the police the gun is Mary's. Mary then objects: "Don't turn this back on me!", as Joe is attempting to redirect the attention of the police from Joe to Mary.