Andoni Iraola's men would break the deadlock through Justin Kluivert before a Callum Hudson-Odoi stunner levelled matters for the visitors, who could not make their extra man count for the final 15 minutes after Billing was dismissed for a horrific challenge on Hudson-Odoi.
While a couple of changes were made on the Cherries' end, as Luis Sinisterra and Kluivert returned to the starting lineup, Forest boss Nuno Espirito Santo switched up over half of his team, and one of his six alterations saw deadline-day signing Matz Sels start in goal.
It was Bournemouth's reintroduced duo who would justify their selections, though, as with just six minutes on the board, a Lewis Cook corner was headed onto the back post by Sinisterra, where Kluivert was waiting to prod the ball home.
Sinisterra wheeled away in celebration before Kluivert was credited with the goal upon review, although Santo's side responded well to the early setback and began to work the gloves of Neto in the Cherries goal.
Taiwo Awoniyi and Ryan Yates both registered shots on target before the half-hour mark for the visitors, who faced another succession of corners in their defensive third, although they dealt with the subsequent set-pieces much better.
Forest continued to huff and puff as half time approached, and their merited equaliser arrived in the first additional minute, as Hudson-Odoi robbed Ryan Christie of possession on the edge of the box, worked the angle for the strike and found the bottom corner with a sublime curler.
A dismal end to the first half for Bournemouth also saw Antoine Semenyo sustain a knock, leading Iraola to bring on Marcus Tavernier for the second period, where Forest dominated the early possession before the hosts regained a measure of control.
Gilt-edged chances were still at a premium for both sides as the clock ticked down, but there was time for a dose of late drama, as Philip Billing caught the back of Hudson-Odoi's leg with a shocking challenge and was given his marching orders.
The onus was certainly on Forest to make the most of their extra man in the short time that remained, but Hudson-Odoi was unable to play on after Billing's tackle and made way for debutant Rodrigo Ribeiro.
Seven minutes of injury time was not enough for Santo's side to find a winner, although they rise back above Luton Town into 16th place by virtue of their point, while Bournemouth also climb one spot into 12th, usurping Fulham.
Bournemouth have five full days to recuperate before travelling to Fulham on Saturday, but there is no respite for Forest, who host Bristol City in an FA Cup fourth-round replay on Thursday.