That’s a surprisingly tricky question; let me start with what is hopefully a helpful analogy. Volcanoes aren’t like old people who might retire from their job or kick the bucket at a predictable age. Volcanoes are much more like a rock band. Most are one-hit wonders, many produce some great hits for a while and then slowly disappear, and others keep rockin’ on for ages.
So how long a volcano remains active for or how long before it’s considered extinct depends on the type. It’s the Rolling Stones vs. Los Del Rio.
Alright, let’s sprinkle a little science in there. Your Los Del Rio or Baha Boys one-hit wonders, in the volcano world, are called monogenetic volcanoes, usually appearing in fields intuitively called “monogenetic fields.” If you consider that genetic means origin and mono means one, then that name is really about as close to “one-hit wonder” as you’re ever going to get in science. They erupt maybe for a few weeks, maybe for a few years, and then they’re done. Forever. They released their big single and now they’ve got nothing left in the tank. Boom, done, extinct.
This is what a field full of monogenetic volcanoes looks like in Arizona:
Polygenetic volcanoes are probably what you’re more familiar with. These are bigger and erupt multiple times (poly meaning many). Some erupt infrequently every few hundred or thousand or tens of thousands… or hundreds of thousands of years. Others like Kilauea never seem to stop touring with their favorite hits:
Woo! Encore! One more!
What really determines whether or not a volcano is active, dormant or extinct is its plumbing system. I suppose in this way; it really is like an old person.
The plumbing system refers to the (usually) complex network of conduits and chambers below a volcano through which magma flows, usually toward the surface. If the magma is flowing to the surface, then the volcano above it is active. The magma can stop flowing for a while, but still remain liquid down below, possibly making another push to the surface at some point in the future. In that case, it’s dormant. Once everything hardens up and the plumbing system cools and turns to stone, nothing else is going to be pushing its way to the surface and the volcano is considered extinct.
Wikipedia tells me a volcano is considered extinct when it hasn’t erupted in 10,000 years. This might be a decent guideline in many cases, but it’s not a good hard rule. Yellowstone is definitely not extinct, even though it’s been 70,000 years since its last lava flow and 630,000 years since its last major eruption. A volcano in a monogenetic field can relatively safely be assumed to be extinct within one’s own lifetime if it doesn’t erupt again. No need to wait 10,000 years to close the book on that.
So in answer to your question, volcanoes can live to almost any age as long as their plumbing system continues to be fed by fresh magma. The age at which they become extinct depends entirely on what’s happening below them.