The point takes Alan Pardew's side to 30, one more than United in sixth.Palace went closest to scoring in the first half when Damien Delaney had a shot cleared off the line by Simon Francis.
Fraizer Campbell then shot narrowly over for Palace.
Host Bournemouth, which itself went into the match with three straight victories, had chances through midfielder Eunan O'Kane, while Matt Richie also headed wide inside the penalty area.
Bournemouth is now 14th with 20 points.
Related story
Late Stanislas penalty gives Bournemouth win over Palace
