The French captain sat out the whole match from the bench after breaking his nose in France's opening 1-0 win over Austria just four days ago.
Xavi Simons had a goal controversially ruled out for the Dutch, while Antoine Griezmann wasted two glorious chances for France.
A share of the spoils leaves the sides level on four points at the top of Group D, one ahead of Austria who beat Poland 3-1 earlier on Friday.
France take on the Poles, while the Netherlands face Austria in the final games of the section on Tuesday.
Les Bleus had won both meetings when these sides met in qualifying but Mbappe scored four of France's six goals in those games.
Roared on by an orange wall around two-thirds of the Leipzig Stadium, the Dutch nearly pounced on any French uncertainty within a minute.
The explosive pace of Jeremie Frimpong took him clear on goal but the Bayer Leverkusen man could not produce a clean finish and Mike Maignan turned his shot behind.
Aurelien Tchouameni replaced Mbappe in France's only change as coach Didier Deschamps sought the security of an extra midfielder.
Without their captain and talisman, the 2022 World Cup finalists turned to Griezmann for inspiration.
The stand-in skipper stung the palms of Bart Verbruggen from distance.
But the Atletico Madrid forward had a moment to forget with the biggest chance to open the scoring before half-time.
An intricate move played in Adrien Rabiot but when he decided to pass rather than shoot, Griezmann failed to connect on his weaker right foot.
At the other end, Maignan had to produce another fine save low to his left to turn Cody Gakpo's drive behind.
France had the better chances in the second period.
Marcus Thuram fizzed a shot just wide from the edge of the area before Tchouameni could not keep a header down.
The normally reliable Griezmann was guilty of another big miss when his weak effort was turned behind by Verbruggen with the goal gaping.
But the major talking point of the first goalless draw of the tournament saw the Dutch denied a winner.
Paris Saint-Germain midfielder Simons fired in on the rebound after Maignan saved from Memphis Depay 18 minutes from time.
The goal was ruled out by the officials on the field for offside as Denzel Dumfries was adjudged to be interfering with Maignan's ability to stop the shot.
After a lengthy VAR check, which has been a welcome rarity at the tournament so far, the offside was confirmed and France handed a reprieve.
Deschamps rejected the chance to risk Mbappe for the closing stages as he instead sent on Olivier Giroud and Kingsley Coman in search of a winner.
But both sides settled for a point that is already likely enough to secure their passage to the knockout rounds thanks to the safety net that sees the four best third-placed finishers advance.
