The result in Reykjavik halted France's perfect start in European qualifying after three straight victories. The two-time world champions could have sealed qualification if they had beaten Iceland on Monday and Ukraine failed to defeat Azerbaijan.
Runners-up at the 2022 World Cup, France still lead Group D with 10 points, three more than Ukraine, who won 2-1.
Les Bleus were without captain Kylian Mbappe, who scored in the 3-0 victory over Azerbaijan on Friday but limped off the field with an ankle injury.
Iceland took the lead in the 39th minute but the visitors rallied with Christopher Nkunku scoring in the 63rd minute and Jean-Philippe Mateta in the 68th.
The hosts levelled a couple of minutes later to move to four points after four matches, three more than Azerbaijan.
Switzerland also had a chance to secure a World Cup spot but couldn't manage more than 0-0 at Slovenia.
The Swiss would have qualified with victory and Kosovo failed to defeat Sweden. Kosovo won 1-0 to reach seven points in Group B, three fewer than Switzerland. Slovenia have three points and Sweden, also winless like Slovenia, one.
Liverpool's Alexander Isak spurned three dangerous chances for hosts Sweden.
Elsewhere, Germany beat Northern Ireland 1-0 in Belfast to stay even with Slovakia, who beat Luxembourg 2-0, with nine points each atop Group A.
North Macedonia drew with Kazakhstan 1-1, while Belgium came from behind to win at Wales 4-2.
The Belgians lead Group J with 14 points, one more than North Macedonia and three more than Wales.
Europe's 12 group winners qualify automatically, with the 12 runners-up entering a playoff along with the four best-ranked group winners of the 2024-25 Nations League.
The 16 teams will be drawn into four playoff paths, with four teams in each, to determine the four remaining European qualifiers. Playoff matches will be played in single-leg semi-finals followed by single-leg finals within the same international window in March 2026.
The World Cup will be in the United States, Canada and Mexico and is the first with an expanded field of 48 teams, up from 32.
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