Nuno Espirito Santo's men have spent most of the year in the league top three, but a run of one win in four games has dropped them behind Chelsea, Newcastle and a resurgent Manchester City.
A win at Selhurst Park on Monday (Tuesday AEST) would have brought them equal with Chelsea and Newcastle, but one point means they stay in sixth place, with the top five qualifying for next season's Champions League.
Neither team was able to take control of a sometimes scrappy match, in which eight names went into the referee's book.
Palace took the lead with a penalty kick after an hour. After a video review, Matz Sels was adjudged to have felled Tyrick Mitchell, and Eberechi Eze made no mistake from the spot.
Forest bounced right back within four minutes, Murillo doing enough to deflect a goal-bound shot from Neco Williams away from the keeper.
Eze hitting the woodwork in the dying seconds and Eddie Nketiah having a goal disallowed for offside in stoppage time meant FA Cup finalists Palace have not won any of their past five league games.
"We were unlucky in a few situations, hitting the post, the offside call, but they fought, and many situations we created," Palace coach Oliver Glasner said.
"It's not easy against a team like Forest, so overall I'm really pleased with the performance today."
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