I'll do these out of order.
- Michael is Production Technology Manager of the facility
This is the most standard way for a job title to be. He is the facility power technology manager. Normally you would probably use "the" before the word "Production" here.
- Michael is Production Technology Manager for the facility
This version is reasonably common but takes a little different perspective. He works for the facility, rather than for the company that owns it. Normally there is no difference between these two, but this kind of feels more like you are describing what he does than giving a standard title. Interchangeable with the above version in pretty much all cases.
- Michael is Production Technology Manager with the facility
This is an unusual way to put things. Usually if you are "with" something, it is some organization other than the company you work for. You might possibly use this phrasing if you were a news reporter and he is speaking for the company. It emphasizes that he is allied with the organization, rather than simply giving his title.
You might also use "with" a company if you are speaking of someone who is changing jobs, but this is not a job title
He was with Goldman Sachs but now he's with a local bank.