James Perch looked as though he had snatched a winner for Forest, with a superb overhead kick, only for the keeper to acrobatically palm the ball over the bar and earn the Cherries a vital point to end fears of relegation.
Steve Fletcher had put the visitors ahead against the run of play when he glanced home a close-range header in the 47th minute.
Nathan Tyson swept home Perch's mis-hit shot to pull Forest level nine minutes later, before the home side launched a barrage of pressure.
But Bournemouth - and Stewart - held on stubbornly for a draw, leaving Forest needing victory in the final game of the season at Bradford, to have any chance of making the top six.
Forest could have snatched the lead in the first minute as Tyson sent Jack Lester clear, only for keeper Stewart to beat the striker to the ball and make a brave claim at his feet.
But, from that point on, the home side were largely frustrated by a Bournemouth side intent on giving little away.
It was clear to see why the visitors had drawn five of their previous seven games, as they showed a great defensive resolve, fielding a three man central defence and just a single striker in the form of target man Fletcher.
Only Tyson and Lester really looked as though they boasted the required guile to find a hole in the Cherries defence.
Lester carved out Forest's best opportunity, as his clever pass sent Tyson clear through the right hand channel. The striker forced a fine save from Stewart and, unfortunately for Forest, there was no Red shirt there to benefit as the ball rebounded loose around the penalty spot.
Tyson also turned provider as his turn and pass picked out Nicky Southall's run into the box, from where the midfielder struck a low shot that Stewart did well to palm around the post.
Bournemouth may have been defensively solid, but they provided little attacking threat, with only a single real effort of note in the first half, when James Hayter produced a neat turn and a rising shot that flew over from just inside the box.
That made little difference as they snatched the lead from a set-piece two minutes after the interval.
Stephen Cooke's delivery from a free-kick on the left was inch perfect and Fletcher, who missed much of the season through injury, flicked home an angled header from eight yards.
The goal sparked Forest into life however, and it was not long before they struck an equaliser, as Kris Commons broke powerfully down the left and into the box, before cutting a dangerous cross back across the box.
Perch picked up the ball and didn't seem to connect properly with an 18-yard shot, but it fell perfectly for Tyson, who side-footed home from close range.
Forest stepped up the pressure as they searched for a winner and began to pick a few holes in the Bournemouth defence - although keeper Stewart held firm.
Julian Bennett broke superbly down the left and crossed perfectly to the edge of the six-yard box, where Perch connected with a spectacular overhead kick - only to be denied by a breath-taking save from Stewart, who somehow managed to palm it over the bar.
Wes Morgan also had a close-range effort ruled out for offside, as Bournemouth hung on for a point that left Forest needing a win at Bradford next Saturday to have any chance of making the play-offs.