Well, I'm familiar with tintypes and have a couple that have come down the family, but this is a new expression to me. I Googled it and found it quite common but seemingly only on US sites, I found three related usages, also on US sites:

“Mull that over on your tintype, then, as we go forward.”

“What did the gladiator say? Blow it out your tintype. Finbad the Failer lives!”

“…or ninny on your tintype, as people of Coppola's father's generation used to say…”

I don’t read much into these as I think they are examples of people not understanding the reference and consequently misusing the expression.

Two possibilities for the origin come to mind.

One is that tintypes were originally rare, expensive and much prized (although eventually very common) and the thought was “Not on your life…not even on your tintype!” I think this is quite unlikely to be the origin, but offer it for what it’s worth – not a lot.

The other is that tintypes were seen as true impressions of reality (the camera never lies – ha, ha) so that “Not on your tintype!” implied something that was *unreal. I think this is quite a feasible explanation.