Herbert, the 2020 Omega Dubai Desert Classic winner, backed up opening efforts of 64 and 67 with a 70 to move to 15-under-par, leading by one from Johannes Veerman after the American carded a 67.
On a disappointing day for former winners Rory McIlroy and Shane Lowry Forrest was just three shots off the lead before a closing double-bogey 6 dropped him into a share of seventh spot with Ramsay on 10-under.
“It would be a pretty cool one to achieve,” said Herbert of setting up a chance of recording a wire-to-wire win. “To win the Irish Open would be really cool, whether it's wire-to-wire or not.”
While pleased to be out in front, the 25-year-old was slightly disappointed that two bogeys on the back nine after bagging four birdies going out had left him with the narrowest of cushions.
“In one aspect, it's kind of frustrating,” he admitted. “I could have really put a good one away and felt like I got too far away from the majority of the field, but 15-under has probably just left the door open a touch. So still going to have to really be on it tomorrow.
“If I can get to 20-under, I feel like it's going to be pretty hard to catch, and if I am caught, someone has played really good golf and they probably deserve to win it.”
Bidding for his breakthrough European Tour victory, Forrest had battled hard to get into red figures for the day before having to settle for a 73.
In contrast to his finish, three-time tour winner Ramsay covered the last five holes in two-under as he signed for a 70.
The pair are still in with a chance, though, and are also in the thick of the battle for three spots up for grabs in the 149th Open at Royal St George’s.
The spring in McIlroy’s stop following a second-round 67 disappeared after the 2016 winner followed a bogey at the third with a double-bogey at the next hole.
Four birdies in the next nine holes was more like it from the four-time major champion before running up another double-bogey 6 at the 16th. A 73 for four-under left him in a tie for 48th.
“I got off to a rough start, did well to claw myself back into it then I hit one out of bounds on 16,” he said in summing up the effort. “Two bad tee shots, one on four and one on 16, cost me four shots today.
“That wasn't ideal but, apart from that, the rest of the round was actually okay. Hit good putts but they just weren't going in. So just one of those days.”
McIlroy is also playing in next week’s Scottish Open and said of that: “It's good I added that in and it's another competitive start.”
Marc Warren (70), Connor Syme (71) and David Law (72) are all a shot ahead of McIlroy, but a 77 left Scott Jamieson on two-over.
After a “struggle” as he carded a 74, Open champion Lowry is on three-under.