Adding to the other answers here, assuming you mean dying of “old age” not predation or accidents or illness, lobsters are supposed to be “functionally immortal” but as they get older, it gets harder and harder for them to molt. If the molt takes too long they will die mid molt.
“According to Carl Wilson, lead lobster biologist with the Maine Department of Marine Resources, between 10 and 15 percent of lobsters die naturally each year as they shed their exoskeletons because the exertion proves to be too much. Each molting process requires more and more energy than the one before it as lobsters grow in size.
“Finally, older crustaceans stop shedding their exoskeletons altogether—a clue that they’re near the end of their lifespans. They run out of metabolic energy to molt, and their worn-and-torn shells contract bacterial infections that weaken them. Shell disease, in which bacteria seeps into lobster shells and forms scar tissue, adheres the crustaceans’ bodies to their shells. The lobster, attempting to molt, gets stuck and dies. The disease also makes lobsters susceptible to other ailments, and in extreme cases, the entire shell can rot, killing the animal inside.”
‘“Is that senescence? Maybe not in how we think about it,” says Jeffrey D. Shields, a marine science professor in the Virginia Institute of Marine Science of the College of William & Mary. “But it is senescence in the way that older people die of pneumonia.
How large could a lobster grow, if it could keep going through difficult molts?
However that leads to an interesting side question I think, for those of us interested in xenobiology and the possible biology of extraterrestrials. What if you had a civilization of lobster like creatures? They would be able to cure the diseases and they could invent machines to speed up the molt, so that they don’t die mid molt. Would they be able to live for ever?
I expect extra terrestrials with technology would develop the ability to extend their lives way beyond what is normal - we do already by a decade or so, average lifespan anyway if perhaps not so much for the maximum lifespan. But some might be able to do that more easily than others. Might it be that lobster type ETs, achieve immensely long lives, at an earlier technological stage than most? Certainly it would be a top priority to deal with that issue of 10 - 15% of your population dying each year mid molt.
In the case of crustaceans like the lobster there’s another issue, “squishy inside” modern arthropods (creatures with an exoskeleton) never get quite as large as the largest “squishy outside” creatures like elephants and especially blue whales. arthropod | animal phylum And the lobster keeps getting larger as it grows older. So eventually surely it will hit a limit of size. It gets oxygen through tiny holes in its skeleton - and as it gets larger, its exoskeleton’s area goes up as the square of its length but its insides volume go up as the cube. So it’s increasingly hard to get enough oxygen trough the holes in its exoskeleton to supply its body.
With animals with lungs the capability to get oxygen goes up nearly as the cube like their volume, because our lungs are filled with fractal structures that expose an immense surface area to the atmosphere inside our lungs. So this not such an issue for us.
So arthropods to get arbitrarily large they’d need to develop lungs, or fractal structures of some sort to be able to expose enough surface area to get enough oxygen into their bodies to keep going. See also Could Insects Reach the Size of Humans? which mentions some other potential issues with immensely large arthoropods.
In the past some arthoropods actually got larger than humans, in the sea, but that may be due to higher levels of oxygen in the atmosphere.
Fossil of world's biggest bug found
Fossil of Jaekelopterus rhenaniae - this ancient species of fresh water “sea scorpion”, Jaekelopterus, grew to 2.5 meters in length (8 feet). May be due to higher levels of oxygen.
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