How come a comet never dies out?

They do die out eventually. Everytime they come close enough to the sun their surface evaporates at an enormous rate. Comets are pretty big however and their orbits are very long so it takes a long time for them to disappear completely. Unless they crash into the sun that is. That is actually quite common and when a comet does so it´s gone forever. Comets can also crash into other bodies in the solarsystem although this is not as common.

Comets doesn´t burn. What you see is the gas and the dust that is boiling off of the comets surface. It shines from sunlight hitting it and by charged particles from the sun interacting with it. This dust remains in the same orbit around the sun as the comet itself and when earth sometimes passes through such an orbit we can see the dust as a meteor shower. Theres one called the Lyrids coming up in late april.

the tail doesn't actually burn. It is just a trail of gas and dust that is left behind.

they do, they just live a hell lot longer than you