Ghana were left without an opponent for the scheduled second match after Togo pulled out of the competition following the deadly attack on their team bus last Friday.
Ivory Coast, Africa's leading contenders at this year's World Cup in South Africa, enjoyed most of the possession but Bakary Kone failed to convert three clear chances and Didier Drogba was far from his brilliant best.
"It was a difficult start. After what happened a few days ago, it was difficult to concentrate," Drogba told reporters.
"We do not forget what happened (to Togo) but now the competition has started and we try to focus on the games."
Ivory Coast coach Vahid Halilhodzic said luck had not been on his team's side.
"We had a lot of chances against an opponent who had decided to refuse to play," the Bosnian told a news conference, adding he feared his team would concede a last-gasp goal.
"We could have lost. On a counter attack, you can concede a goal. They can shoot once and score. We lacked a bit of luck. I am disappointed but it's not over yet."
Burkina Faso coach Paulo Duarte told a news conference: "We were not good in the first half, we did not create chances. It was better after the break."
Ivory Coast now face the prospect of early elimination as they have only one group game left, against Ghana, on Friday.
Burkina Faso will take on Ghana on January 19.
MINUTE'S SILENCE
There was a minute's silence before kick-off in memory of the Togo assistant coach and the press officer, who were killed in an attack on their bus by a separatist group.
Togo left Cabinda on Sunday, pulling out of the tournament and leaving Ghana without an opponent for Monday's second game in Group B scheduled for 6:30pm GMT.
Security was reinforced around the Chiazi National stadium with police officers searching cars in front of the entrance and a police helicopter with armed officers on board circled above the area before kick-off.
Ivory Coast got off to a good start and were denied a third-minute penalty when Drogba was brought down in the box.
A Kone 20-metre drive then went just over the bar as Burkina Faso struggled.
Yaya Toure's header was cleared off his line by a defender shortly before the half hour with Kone again being denied by Burkina keeper Daouda Diakite on the stroke of halftime.
Marseille striker Kone missed the target 10 minutes into the second half with a fine shot from just outside the box.
Burkina Faso improved after the break while the pitch and Ivory Coast deteriorated.
Moumouni Dagano forced Boubacar Barry to make a spectacular dive with a powerful 40-metre free kick and Kone then missed the open goal after being perfectly set up by Gervinho in the 69th minute.
He was immediately replaced by Abdul Kader Keita.
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