During an opening round in which many of the game's superstars had trouble with the Pete Dye-designed TPC-Sawgrass Stadium Course, Poulter did not miss a putt inside 10 feet and Laird birdied all four of the par-5 holes.
They had a one-shot lead over Blake Adams, a second-year PGA Tour player who has not come close to winning on the circuit, and a two-shot advantage over Kevin Na and Ben Crane.
Na's chance to at least share the lead was wiped out when his tee shot at the first hole stuck high in a palm tree, resulting in a double bogey.
None of the top 11 ranked players in the world managed to break 70, and of the first 16 on the leaderboard, there was only one major championship winner -- former British Open champ Ben Curtis.
Curtis, who ended a six-year victory drought last month with a win in Texas, was part of the large group tied for sixth at 68 that included Matt Kuchar, Jhonattan Vegas, Kevin Stadler, former Players champion Adam Scott and last year's FedEx Cup winner Bill Haas.
Most of the game's big names dug themselves a hole in the opening round.
Mickelson birdied the first two holes and then struggled the rest of the way. His round was symbolized by his ability to save par at the par-5 ninth after hitting his tee shot in the water.
McIlroy had a chance for a solid first round until he put his tee shot in the water at the famed par-3 17th with its island green.
Hunter Mahan, the only two-time winner on the tour this year, stumbled to a 74. That was the same score turned in by Tiger Woods, who will begin Friday's round trailing 99 players and facing the possibility of missing the cut for the second straight week.