The Jargon File (version 4.4.7, 29 Dec 2003):

flamer
 n.

    [common] One who habitually flames. Said esp. of obnoxious Usenet
    personalities.


The Free On-line Dictionary of Computing (30 December 2018):

flamer
pain in the net

    (Or "pain in the net") One who habitually
   flames.  Said especially of obnoxious Usenet
   personalities.

   [Jargon File]

   (1996-08-26)