Ahead most of the day, Poulter hit two balls in the water on consecutive holes to fall into a tie for the lead with Casey at seven under par.
Play was suspended because of darkness at 6:18 p.m. EST. Poulter has 11 holes remaining and Casey has nine; play resumes at 8 a.m. EST Monday.
Patrick Reed, who briefly held the lead at seven under after Poulter's misadventures, was third at six under. Daniel Berger, who is playing in his first Honda, holed out for birdie from 21 yards on the 11th hole to move into a tie for fourth at four under with Phil Mickelson and three others.
"I played wonderful golf for my first 18," said Casey, who shot a two-under-par 68 to get to 3 under for the tournament. "My third round today, tee to green, was fantastic. I didn't hole the putts I wanted to."
Poulter shanked his eight-iron tee shot into the water on the 174-yard, par-three fifth hole and then hit his drive in the water on the par-four sixth to lose what had appeared to be a pretty solid lead.
The wayward shots resulted in a double-bogey and a bogey that dropped Poulter a stroke behind Reed. Poulter bounced back with a birdie on the par-three seventh after hitting his tee shot to within three feet. Reed missed the green and failed to get up and down.
"Look what happens when you forget to concentrate," Poulter said. "Silly things happen.
"I tried to take too much off an eight-iron and hit a beautiful shank. So it was an easy double bogey and poor tee shot on the following hole. But I'm pretty pleased with the golf I've played throughout the whole of today. I haven't really made many mistakes at all. I've put it in position an awful lot, which is encouraging, and if I do that tomorrow, then I'm going to be in a good position."
A fierce thunderstorm Saturday swamped the Champion course at PGA National Resort & Spa and stopped third-round play. Thanks to a Herculean effort by the grounds crew, the course was made playable and the third round was resumed at 10 a.m. on a sunny, breezy Sunday, with players allowed to lift, clean and replace the ball in the fairway.
When that round ended, Poulter's four-under 66 had given him a three-shot lead over second-round leader Padraig Harrington and Reed and a six-shot lead over Casey.
Casey, who was three groups ahead of the leader, birdied three of his first four holes in the fourth round, which started at 2:40 p.m. A birdie at No. 9 got him to seven under.
"Right off the blocks I made the putts when I needed to and the putter seems to be working nicely, which is the difference why I've snuck up on the leaderboard," said Casey, who explained that his late afternoon putts fell because of a change in green-reading strategy.
"I struggled with the putting for my third round and I felt like I wasn't reading the grain enough on the putts. So this afternoon, I've played more of the grain to the way the putt is going to break and less of the slope. That's been the difference."
Harrington began the day at seven under, one shot ahead of Reed and two shots ahead of Poulter.
Poulter birdied the first hole and was tied for the lead after Harrington bogeyed the second. Poulter added three more birdies to get to nine under and Harrington was at six under after a 71. Reed shot 70 to also finish at six under.
During his third and fourth rounds, Poulter made 10 straight pars to keep his lead while Reed never got closer than three shots and Harrington fell four behind.
While Poulter was double-bogeying the fifth, Reed rolled in a 34-foot birdie putt from off the green to tie Poulter at seven under. When Poulter bogeyed No. 6, Reed went ahead, albeit briefly.
"Just a loose tee shot," said Reed of his miss left of the hole on the seventh. "I've hit a lot of those today, and I've been scrambling and scrambling. I happened not to get one there, but I've got a lot of holes left and I'm only one back.
"Any time you have a chance to win the tournament, it's always good."