The result eliminated the East Africans, who had needed to win to progress themselves.
A point was enough for DR Congo to finish second in Group F behind Morocco, who topped the table thanks to their 1-0 win over Zambia.
The prize for the Leopards is a last-16 tie against Egypt, with Morocco taking on South Africa.
The results in Group F were also enough to see tournament hosts Ivory Coast sneak into the second round as the last of four best third-placed teams.
The Elephants – who had sacked boss Jean-Louis Gasset earlier on Wednesday after finishing third in Group A – will face defending champions Senegal in the last 16 on Monday.
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Leopards have better of stalemate
In a drab encounter almost entirely lacking in clear chances, neither side truly came close to breaking the deadlock in what was the first Nations Cup finals meeting between the two nations.
Urged on by a vocal, vibrant set of supporters that dwarfed the contingent backing Tanzania, DR Congo created the better opportunities in the opening period.
Gael Kakuta fired a free-kick straight at Taifa Stars goalkeeper Aishi Manula, who was also quickly off his line to block Fiston Mayele’s effort right on the edge of his penalty area.
Kakuta’s clever through ball then picked out the run of Yoane Wissa but the Brentford forward scuffed his shot after cutting inside.
Meschack Elia could not help DR Congo break the deadlock after coming on as a second-half substitute
Tanzania captain Mbwana Samatta had his side’s best effort, spinning in the box to fire over from a tight angle, and chances were at even more of a premium in the second half until Wissa created two openings late on.
First, he brought another good save from the backpedalling Manula, forcing him into a full-length dive to claw away an attempted chip, before blazing over after a scramble from a corner.
But in truth, as a light mist descended on the Amadou Gon Coulibaly Stadium, neither side looked like finding their way to a goal.
The result means Tanzania claim their first ever Nations Cup clean sheet, yet remain winless in nine matches at the tournament, finishing bottom of their group once again.
Two-time winners DR Congo have reached the knockout stage for the fifth time in their past six appearances at the finals and will fancy their chances of overcoming an Egypt side missing Mohamed Salah when the two sides meet in San Pedro on Sunday (20:00 GMT).