How long do tomatoes live for under ideal conditions?
Do different varietals have vastly different lifespans?
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Sue encouraged me to turn my comments into an answer. So, I'll try to make them meet the criteria.
I've only ever heard of tomatoes living a few to several years at the most. Ecnerwal's answer sounds pretty apt. However, I think it's possible for them to live much longer. That's me, though. Whatever the case, vigorous indeterminate tomatoes will probably live the longest. Determinate ones will probably die after they fruit unless you take cuttings of them or something. Indeterminate ones are supposed to grow and fruit indefinitely (until it frosts, if there is frost). Indefinitely is a long time, however. In reality, most indeterminate tomatoes will probably live until they get too diseased or run out of root room for too long, or run out of nutrients, or something. Pruning may or may not help. I'm not sure.
Ideal conditions would require an absence of disease-spreading bugs (e.g. aphids and whiteflies). In some areas, that means indoors. They need plenty of light, plenty of soil, fertilizer, enough calcium and temperature regulation. I find that tomatoes are actually easier to grow indoors in the fall/winter/spring than in summer (it's too consistently hot indoors then, in my main growing environment, but they still do reasonably well in the kitchen windowsill). Frost will kill tomatoes, normally, although some might survive a light frost.
Also, realize that you can take tomato cuttings. So, if it's really old and you want to keep the same plant going, that's one way to rejuvenate it, so-to-speak. It should work for both indeterminate and determinate tomatoes.
Anecdotally, I met a "tomato tree" (not really a tree, just a vary large old tomato plant) that was reported to be 4-6 years old (likely started after the last serious hurricane) on an island in the tropics. The main trunk was over 4" in diameter.
I'd suspect that determinate/indeterminate makes a big difference here.