Wordorigins.org

To fire is an American slang verb meaning to dismiss someone from employment, to sack someone. The underlying metaphor is not quite certain, but it is most likely that of a bullet being fired from a gun.

The word first appears in the early 1870s in a general sense meaning to dismiss, to get rid of someone. From the March 1871 issue of Overland Monthly:

The thought that I was fired by some stranger, who wasn’t a-takin’ no hand...is not a good thought to die on.

The dismissal from employment sense appears a decade or so later. From Sweet and Knox’s Texas Siftings of 1882:

If Gould fires you out, the only railroad in Texas that will employ you will be some street railroad.

There are various claims floating about that the slang term comes from literal acts of arson. An employer would burn the desk or home of someone he wished to dismiss. This is utter nonsense. The slang term is metaphorical and was never meant to be taken literally.

(Sources: Historical Dictionary of American Slang; Oxford English Dictionary, 2nd Edition)