Taylor had scored in each of the Addicks’ first three Sky Bet Championship games and he added a fourth after 18 minutes at The Valley.
The goal came at the end of a move involving 22 passes when Taylor headed Charlton in front but Forest proved resilient and earned a point as Adomah pounced on a corner to shoot home from close range late on.
Charlton started the game on the front foot and Jonathan Leko’s shot came back off the post just two minutes in.
Four minutes later the home side went close again as Aro Muric kept out a shot from Conor Gallagher before Michael Dawson got in the way of an effort by Leko from the rebound.
But there was no stopping Charlton after 18 minutes as Taylor headed in from a Leko cross following a flowing move by the home side.
Forest tried to hit back but Alfa Semedo curled over following a cross by Matty Cash.
Ten minutes before the break Josh Cullen played a one-two with Erhun Oztumer before his effort was cleared by Muric.
Forest’s Lewis Grabban looked set to draw his side level in stoppage time at the end of the first half but Tom Lockyer’s block denied him from close range.
Charlton had the first opening of the second half but Leko scooped over from 12 yards after Taylor did well to cut the ball back.
But Forest responded with a period of pressure which paid off after 78 minutes as Adomah struck the equaliser.
A corner into the Charlton area saw the ball bounce off a defender’s leg before Adomah fired home.
Both sides had late chances to snatch a winner but Jonny Williams saw his deep cross beat Muric and drop just past the post before Forest’s Tiago Silva hit a dipping long-range volley which cleared the bar.