McIlroy, who was playing with Thriston Lawrence and Kristoffer Reitan, was frustrated after the warning at the K Club. The world number two finished the day five shots off the pace.
Dane Thorbjorn Olesen, France's Romain Langasque and Spain's Ignacio Elvira all hit six under-par 66 rounds to share the lead.
"In all honesty, I felt a little rushed out there for the last 12 holes," said the Northern Irishman, a five-time major winner.
"We got put on the clock pretty early, and then the first official went away, and then we were put on the clock for the last three holes to try to make time up."
McIlroy was not helped by the struggles of South African Lawrence (81) and Norway's Reitan (77), who both hit their ball into the water on the seventh hole.
"I felt like I played better than what the score reflects," he said.
"I hit the ball well, I drove the ball well, just had that disappointing finish, bogeying two of the last three holes."
England's Daniel Brown, Austria's Bernd Wiesberger and France's Adrien Saddier each posted rounds of 67.
Irishman Shane Lowry, who, along with McIlroy, is a member of Europe's Ryder Cup team, carded three birdies in his last six holes to stay in touch with a 69 at the 7,441-yard K Club course near Dublin, where par is 72.
Five-time major winner Brooks Koepka matched McIlroy's 71.
European Ryder Cup captain Luke Donald finished his first round with a five-over par 77.