The Court of Arbitration for Sport says it rejected Lynch's challenge against Horse Sport Ireland because it had no authority to hear the case.
Irish officials had nominated Cian O'Connor to replace Lynch getting one of two entries in individual jumping.
Lynch was originally selected then withdrawn after his horse, Lantinus, was disqualified from a recent event on health grounds.
The competition begins on Saturday at Greenwich Park.
O'Connor won gold in show jumping at the 2004 Athens Olympics but was stripped of his medal after his horse tested positive for a banned substance.