Can a berserk unit advance in Advanced Squad Leader?
It can't advance while berserk:


  A15.431: ... nor may it move in the
  APh ...


but, it may lose it's berserk status in the Advancing Fire Phase, and then advance in the Advance Phase:


  A15.46 RETURN TO NORMAL: A berserk unit loses its berserk status and returns
  to Good Order whenever it (or the
  group it attacks with) eliminates all
  (but at least one) Known enemy units
  in its Location with either TPBF
  (halved) during the AFPh or CC, or if
  at the end of a charge there is no
  Known enemy unit in its LOS (15.431).