The Reds are now without a win in their last six Premier League matches, while Forest are still waiting for their first victory under new head coach Oliver Glasner.
Both teams had disappointing results on the opening weekend and were eager to respond. Forest started the brighter of the two sides and caused Liverpool problems early on.
Jeremy Jacquet made an important defensive intervention to stop Igor Jesus, while Dan Ndoye headed wide from a promising position.
Forest eventually took the lead midway through the first half. Morgan Gibbs-White was once again at the centre of the move, laying the ball into Ndoye's path, and the Switzerland international calmly guided his finish into the far corner.
Liverpool responded strongly and Florian Wirtz was at the heart of their best attacking moments.
The German thought he had equalised when he turned and fired into the bottom corner, but the goal was ruled out after Jeremie Frimpong was judged to have been offside in the build-up.
Wirtz then saw a header pushed over by Forest goalkeeper Matz Sels, while Munoz wasted another good opportunity when he curled wide from close range.
Forest remained dangerous on the counter-attack and almost doubled their lead before half-time. Igor Jesus met a cross with a first-time effort, but Alisson Becker produced an excellent save to keep Liverpool in the game.
Liverpool eventually found their equaliser on the hour mark.
Cody Gakpo created the opening from the right, picking out Alexander Isak with a simple pass across goal. The striker was left with the straightforward task of tapping into an empty net.
However, Liverpool's hopes of taking control of the match were quickly damaged.
Forest regained the lead within 10 minutes after substitute Liam Delap took a quick throw-in to release Neco Williams. The former Liverpool player attempted to lift the ball over Alisson and, although his effort went wide, the goalkeeper brought him down.
The referee awarded a penalty and Gibbs-White stepped up to beat Alisson from the spot and put Forest back in front.
Liverpool pushed forward again and found another equaliser with less than 10 minutes remaining.
Munoz made amends for his earlier missed opportunity by scoring his first Liverpool goal, smashing a powerful effort off the crossbar and into the net.
The goal gave Liverpool hope of completing a late turnaround, but they were unable to find a winner.
The draw leaves Liverpool with just two points from their opening two matches and extends their worrying winless league run.
