Reds boss Colin Calderwood saw his side twice denied by the woodwork, with Sammy Clingan hitting the post and the bar.
But, while Forest - who were missing six players through injury - dominated possession, Bournemouth could have snatched victory themselves, with Kevin Bond's side seeing keeper Paul Smith produce fine saves to deny Josh Gowling and Darren Anderton.
Both teams included four players making their debut in their starting eleven, with Neil Lennon, Matt Lockwood and Kelvin Wilson having arrived as new signings for Forest in the summer and youngster Brendan Moloney handed his first start for the Reds.
While Bournemouth included a trio who had signed within the previous 72 hours, in Asmir Begovic, Jason Pearce and Mark Gradel - as well as Jean Christophe on the bench.
It was the visitors' new-look side that settled the quickest, as Garreth O'Connor headed just over a gaping goal after keeper Smith had failed to get any distance on his punch.
Smith soon made amends with a superb save to deny Gowling after the defender connected with a powerful drive on the edge of the box, blocking well with his legs.
While O'Connor was guilty of spurning another chance, failing to properly connect with a shot when well placed 12 yards out.
Scott Dobie, Forest's lone striker, came close with a shot driven into the sidenetting, while Clingan came even closer, with a rocket shot that cannoned back off the post.
Forest twice came close to scoring in the opening ten minutes of the second half, with Wes Morgan glancing a towering header inches wide of the post and Clingan being denied by the woodwork again.
This time it was the bar that the Irishman struck with a rising shot after Dobie's clever knock-down.
Veteran Anderton drew another sharp save from Smith with a low shot, after a rare mistake from Forest's own elder statesman, Lennon.
Kris Commons twice threatened from range, while substitute Junior Agogo came agonisingly close to snatching it for Forest in injury time, with a turn and shot that forced the best from keeper Begovic.