Why do comets appear to have a continuous supply of dust?
The rate of loss of mass from a comet is perhaps surprisingly low. A paper The calculation of $Af\rho$ and mass loss rate for comets gives a rate for a "typical" comet at 1.29AU as 153kg/s. A typical comet has a mass of about $10^{13}kg$. Even if the comet is constantly active (and they are not) it would have enough material for over 2000 years. 

In fact comets are only active when the are near perihelion, and they are only near perihelion for a small fraction of their orbit. Put it together and you have the potential for an active life of hundreds of thousands of years.

Eventually, however, comets do run out of volatile substances, at which point they become inactive, and appear as small dark asteroids. An example of which is the object 2015 TB145, which passed the Earth on Halloween 2015. 

Note: not "oxidising", but "vaporising", the ices in the comet are changing into gas, but there is no burning.