This word, which may also be spelled fease or feaze, means "to bother," "to disconcert," "to disturb," "to disrupt." It is a variation of feeze, an obsolete and dialectal word derived from Old English meaning "to drive away," "to put to flight." At best, faze is an informal word; at worst, it is tiresomely overused in such expressions as "Nothing fazes him." Do not confuse faze with phase, a word with a completely different meaning but the same pronunciation.
Dictionary of problem words and expressions. Harry Shaw. 1975.