The meaning should be obvious. Like saying "not on your life," it referrers to the possibility of a promise made by 'you' : "not even if you promised to give your life."
The idea that a tintype was as valuable as a persons life is rather absurd, even back in the day. People gave and traded tintypes like we might give business cards, so "even if you promised to give your tintype" it's a bit silly, like saying "even if you promised to give your business card." It is a type of American humor that could originally come from an actual spoonerism. But the content of the phrase is that it a rebuff to the person giving the offer, suggesting the offer is bogus or an insult. The offer isn't worth the paper it's printed on, isn't worth a tintype.
The phrase itself has merit as a tintype would be a ready and humorous synonym for worthlessness, but it may have it's origin in comedy.
This type of humor was very popular in low comedy, like vaudeville and burlesque, being the catch phrase of a comedian. These acts would travel to every major city, so the entire nation might be entertained by the same comedian, but the acts were rarely recorded or even written down, so without a direct record, review, journal entry, etc. it can be impossible to trace a phrase like this.
Regardless, these types of phases can be catchy and survive generations if they are clever enough, even though the original comedy is long forgotten. Often a catchy phrase like this would become a cliche or stock line for a certain 'type', in this case an angry man. This phrase may have been the catch phrase of a comedian in a long forgotten burlesque act.
P.S. there is a misconception that tintypes were "rare" or "prized" they were the cheapest form of photography, the Instamatic of the day, used by the lower classes. Albumen, Ambrotypes and Dagarotypes were much more expensive and preferred (Lincoln gave thousands of tintypes away on campaign buttons). As such tintypes were a code word at the time for worthlessness, much like business cards today which are practically forced upon people -something of value only to the giver not the receiver.
Tintypes were popular in the U.S. because it was cheap to establish a studio to make tintypes and for itinerant photographers to pack up a mule team and travel newly developing communities. Tintypes never caught on in Europe in the same way, which explains why the phrase is exclusive to the U.S.