The hosts took the lead after five minutes when Justin Kluivert prodded in at the near post from a corner.
Bournemouth ended with 10 men as Philip Billing was shown a straight red in the 83rd minute for fouling Hudson-Odoi.
The draw means Bournemouth are up to 12th in the table while Forest move a point and a place above Luton into 16th.
The Cherries started the match brightly, winning two corners in the opening five minutes and scoring from the second one through Kluivert's close-range finish.
Despite going behind so early in the contest, Forest stayed in the game and defended doggedly.
Former Chelsea winger Hudson-Odoi equalised with a classy strike at the end of the first half, but neither side could find a winner after the break with few clear-cut chances created.
Subdued Cherries struggle to break down Forest
The stop-start nature of the game played into Forest's hands with Bournemouth unable to capitalise on their fast start.
Dominic Solanke made his 200th Cherries appearance but was unable to mark the milestone with a goal as the visitors' defence kept him quiet for the majority of the contest.
Andoni Iraola's side had nine shots but just two on target as they struggled to hit their free-flowing best - and will see this as a missed opportunity to end a four-game winless run in the top flight.
They are next in league action against Fulham on 10 February (15:00 GMT kick-off).
More to follow