After Saturday's play was wiped out by rain there was no chance of completing the fourth round on Sunday and players will return on Monday for the conclusion.
Poulter's third-round 66 propelled him into the 54-hole lead, with a three-stroke advantage, but he slipped in fourth-round play with a double bogey on the par-three fifth and a bogey on the sixth.
But the Englishman restored his spirits with a birdie on the par-three seventh before darkness ended play.
"I was tiring and made a couple of really bad swings on five and six," said Poulter.
"But that kind of angered me inside enough to spark a little bit of energy there to hit a good shot on seven.
"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 right now, and if I do that on Monday, then I'm going to be in a good position," he said.
Casey revelled in the dusk at PGA National, reaching the turn at four-under, bogey-free, to pull level with his compatriot.
"Sleeping on this sort of situation, is going to be difficult for everybody, it just breaks up momentum," said Casey.
"Some guys will carry it through tomorrow. Others won't and that's very difficult to predict. You just hope you wake up tomorrow and you feel like you've got the same kind of golf swing and the putts are going in the hole. I hope the golfing gods are nice to us."
American Patrick Reed, who shot an even-par 70 in his third round, was in third place, a stroke behind the leading pair.
Overnight leader Padraig Harrington fell to four strokes off the lead after a third-round 71 was undone by a bogey on the fourth and a double bogey on the sixth, his penultimate hole of the day where he got in trouble in the rough.
Five players were three shots off the lead including Phil Mickelson, who had a third-round 69 and was one-under through the first eight holes of his fourth round.
World number one Rory McIlroy had missed the cut after shooting seven-over from his first two rounds.
(Reporting by Simon Evans in Miami,; editing by Justin Palmer/Gene Cherry)
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