The Eagles of Carthage dominated large proportions of the first half, where Issam Jebali had a 24th-minute effort ruled out for offside.
In Christian Eriksen's first major tournament appearance since suffering a cardiac arrest at Euro 2020, Denmark offered little in the opening 45 minutes, but they improved after the interval.
Andreas Skov Olsen's second half strike was chalked off by the assistant's flag, while Denmark looked set to be given a stoppage-time penalty for an apparent handball, but referee Cesar Arturo Ramos decided against awarding the spot kick after consulting VAR.
Despite failing to find the elusive winner, the result ensures that Denmark remain unbeaten against African opposition at the World Cup, while Tunisia will be pleased with their second-ever clean sheet in the tournament's history.
Tunisia began the contest on the front foot, and they caught their European opponents off guard with their approach, as they hustled and harried in a bid to find the opening goal.
The first shot in anger came from the right boot of Mohamed Drager, whose powerful strike took a wicked deflection off Barcelona defender Andreas Christensen before drifting agonisingly wide.
As the contest reached the midway point of the first half, Tunisia fans thought that their team's hard work had paid dividends when Jebali ran beyond the Danish defence to poke the ball past the onrushing Kasper Schmeichel, but the assistant's flag cut the celebrations short.
Touted as a potential dark horse in the tournament by many, Denmark failed to live up to that tag in the first half where a rather tame Pierre-Emile Hojbjerg effort represented their best chance of the opening 45 minutes.
In truth, they were fortunate to head down the tunnel on level terms after Aissa Laidouni fired the ball over the bar, while Schmeichel produced a stunning save to swat away Jebali's delicate chip, which may well have been ruled out for offside.
Denmark had to negotiate another period of Tunisian pressure after the interval, with some impressive last-ditch defending ensuring that the score remained goalless.
However, it was the European nation that had the ball in the net in the 55th minute when the ball broke to Skov Olsen, who smartly guided his shot into the bottom corner, but like Tunisia in the first half, Denmark saw a potential opening goal cancelled out by the assistant's flag after Mikkel Damsgaard had taken up an offside position in the build-up.
Despite seeing Skov Olsen's effort chalked off, Denmark continued their determined pursuit of an opener, as Eriksen produced a menacing strike that forced Dahmen to smartly tip the ball over the crossbar.
From the resulting corner, Mathias Jensen rose highest to accurately nod the ball into the path of Andreas Cornelius, who spurned a glorious opportunity, flicking a header onto the post from close range.
Tunisia's attacking threat diminished as the second half wore on, but they may have felt hard done by to see a penalty appeal waived off after the ball struck the outstretched arm of Crystal Palace's Andersen.
After surviving Tunisian's vociferous calls for a spot kick, Denmark unleashed an onslaught in the final stages, and their pressure looked set to be rewarded with a stoppage-time spot kick when the referee was called to the VAR monitor for a potential handball against Yassine Meriah, however, to the surprise of many in the Education City Stadium, the Mexican official decided against pointing to the spot.
The referee's decision ensured that the spoils were shared, with Denmark now set to turn their attention to Saturday's meeting with the defending champions France, while Tunisia will face Australia.