But there was late drama as the two-time US Open champion Bryson DeChambeau was hit with a two-shot penalty by the R&A after being adjudged to have inadvertently improved the area of his intended swing by treading down long grass after finding rough at the fifth.
DeChambeau was seen in animated discussion with rules officials as he was driven back in a buggy to the scene of the incident where his ball had ended up in long grass.
Later he emerged from a cabin and headed to the range without speaking to reporters other than to ask "Are you guys having a great night?".
Prior to that, DeChambeau ended his run of missed cuts at majors this year in style with a birdie-birdie finish for a 66, and will start the third round three shots off the lead.
World number 97 Herbert doubled over his putter in anguish as a short attempt for par on the 18th slipped by with the chance to become the first man to post a major round of 61.
Burns followed suit less than an hour later after six birdies on the back nine, including a chip-in from the bunker on 18, to move to five under for the tournament.
Under glorious sunshine in Southport on England's north-west coast, Herbert got off to a flying start with birdies on his opening three holes and added three more at the fifth, seventh and ninth to hit the turn in 28, a British Open record for the front nine.
His momentum continued with birdies at the 12th and 13th before a perfect approach set up another three at the par-four 16th for a shot at history.
Despite a wayward tee shot at the last, he should still have scrambled for a par, but the moment in front of huge crowds seemed to get to the 30-year-old as two tentative putts resulted in his first bogey of the day.
"It was not lost on me the amount of history in major championships and the opportunity I had to obviously break the record," said Herbert, who has spent the last two years on the breakaway LIV Tour.
"To tie it as well is still something I'm really proud of."
Burns took the opposite route to 62 with a six-under 30 on the back nine to surge into contention.
The world number 18 sank a 20-foot birdie at the 17th and then chipped in from the greenside at the last for his share of history.
"I caught myself by surprise," said Burns. "I think the finish there the last three holes was just a bonus."
Both players equal the course record set by Branden Grace the last time the British Open was held at Birkdale in 2017, one of five previous rounds of 62 at majors.
McIlroy makes the cut
Two-time US Open champion DeChambeau had missed the cut in all three previous majors this year and has traditionally struggled at the British Open, but is making the most of the benign weather conditions.
The American outshone playing partner Scottie Scheffler to ensure he starts in the final group with Herbert on Saturday.
World number four Cameron Young is well in the running for his first major at six under par, alongside overnight leader Jackson Suber.
Defending champion Scheffler is four shots back but could easily have been at least in a share of the lead but for a series of missed short putts with birdie opportunities for the world number one.
Rory McIlroy recovered from a difficult opening round of 72, which included six bogeys, to at least put himself in position to mount a surge up the leaderboard over the weekend
The six-time major champion posted a 67 to move to one under for the tournament.
"I think the main objective today was to be here for the weekend, which I am," said McIlroy. "I think if I can get off to a decent start tomorrow, be four of five under for the tournament, I'll be right in it."
Tommy Fleetwood's late charge was met with deafening roars around his home town with three birdies in the final four holes to get to four under par.
World number three Matt Fitzpatrick and Justin Rose were among the big names to miss the cut.