The two sides were due to meet at Windsor Park in the third round of the competition, however the Northern Ireland Football League confirmed in a statement the game would no longer be going ahead.
NIFL confirmed the player in question had played in a previous round of the competition.
NIFL said in a statement: "Following communication from the Irish FA Player Registration Committee today (Tuesday), the NI Football League have been made aware of a player eligibility issue relating to an Institute FC player.
"As the player concerned has played in the previous round of the BetMcLean League Cup the NI Football League can confirm that the third round tie between Linfield and Institute (scheduled for Wednesday 13 October 2021) has been postponed until further notice."
'Stute beat Premier Intermediate League side PSNI 4-3 in the second round on 14 September.
Four third round ties will proceed as planned on Wednesday night: Carrick Rangers v Coleraine; Crusaders v Ballymena United; Glenavon v Glentoran; Warrenpoint Town v Loughgall.
Two remaining ties - Portadown v Ballinamallard and Ards v Cliftonville - will be played on Tuesday 26 October.