

There are a couple definitive sources for Irish surnames -- Griffin and Grenham for more recent periods, the Conradh na Gaeilge compiled database and DCU's Bunachar for all recorded history -- and none of them include Pheysey.
Bardsley's 1896 Dictionary of English and Welsh Surnames says:
This surname is derived from a geographical locality. 'de Veci' or Vesci. Lower says,' Robert de Veci assisted William I to conquer England, and was rewarded with great estates in the counties of Northampton, Leicester, Warwick, and Lincoln. Ivo or John de Veschi was his near kinsman, and from him in the female line descended Lord Vesey' (Kelham's Domesday); v. Patronymica Britannica.
Willelmus de Vesci, 7 Henry II: Pipe Roll.
Eustace de Vesey, Lincolnshire, 1273. Hundred Rolls.
Richard de Vesey, Yorkshire, ibid.
1512. John Veysey or Vesey, or Voysye, or Pheysy: Register of the University of Oxford.
1603. Walter Veysey, Devon: ibid.
1603-4. James Voyzey, Devon: ibid.
Other spellings of the names of the two students last mentioned are Voysey, Vesey, and Veisey (v. Index).
So that probably rules out a connection to the surname Fish as well.