What propels comets on their travels and why don't they ever burn up?

Comets are propelled by the gravitational forces of other celestial bodies around them, and they don't burn- they melt. They can eventually lose enough icy material that they no longer generate a tail, at which point the nucleus that remains becomes an asteroid, rather than a comet.

Current theory proposes that most comets originate in the Oort Cloud - the vast cloud of comets that surrounds the solar system out to an estimated distance of two light-years - or the Kuiper Belt - the belt of comets and asteroids extending from between 30-50 A.U.. Comets usually stay in the Oort Cloud or Kuiper Belt; however, if a passing object (such as a star) exerts enough gravity, the comet might be nudged free. Sometimes its new path will expel it from the solar system all together, and it is doomed to roam interstellar space. The alternate path takes it into the inner solar system, where it can become a short- or long-period comet.

As it travels towards the sun, the gravitational pull of the planets changes its trajectory. About the time it passes through Jupiter's orbit, the nucleus begins to feel the heat and pressure of the sun, and surface ice starts to melt, forming the extra features that make it recognizable as a comet. The first encounter with the sun usually changes its orbit the most. … Due to the nature of how the tails form -- material being blown off of the comet - the tails are how comets loose the bulk of their mass. The material usually dissipates after several hundred years, but before that happens, the material usually will continue in the orbit of the comet. If the Earth plows through this, we see a meteor shower.

http://jtg.sjrdesign.net/comets.html

A factor effecting the lifetime and why comets are slow to burn up has to do with gravity also. When those near Sol particles are driven off by Sol's radiation, they aren't left in place. Celestial mechanics tells us they travel along in the same orbital path as the main comet body (much like the rings around Saturn only on a much less dense concentration and a vastly greater circumference). Once out of the inner solar system, the particles can be re-attracted to the main comet body, re-establishing its mass. It will also encounter other new material in its travels building its size again until its next Sol near death experience. Being relatively lighter and more abundant than other molecules, water ice will be one of the easier to attract particles involved in that rebuilding.

https://en.wikipedia.org/wiki/Gravity