Yes, but you could very well think about something equivalent to the time-turner that exchange matter between the past and the present, instead of just sending matter to the past, in a way that keeps energy conservation. It would be harder to use practically,but wouldn't change anything to "energy conservation" vs "determinism" issues.
Don't forget that to fully conserve energy, you have to maintain not only the total mass, but also the internal chemical potentials of whatever thing you're shifting into the past and its gravitational potential energy with respect to the rest of the universe. I think you'll have a hard time doing this without just making an exact copy of the object. "Conservation of energy" is a much harder constraint than is obvious from the three words of the English phrase.
This is a thread where people can ask questions that they would ordinarily feel embarrassed for not knowing the answer to. The previous thread is at close to 500 comments.