Though “the Old Forest” was totally left out in the movie, there is some sort of cameo in TTT EE.

What happens in the movie:
When Merry awakes in Fangorn, he sees Pippin drinking water. After the little “argument”, the Hobbits are caught by a big tree. Then Treabeard appears and says “Away with you. You should not be waking. Eat earth, dig deep, drink water, go to sleep. Away with you.” This scene surely refers to the lines of Tom Bombadil.

What happened in the book
There were also Merry and Pippin caught, while Frodo and Sam were crying and looking for help.
Here is a bit of it:

Master Merry’s being squeezed in a crack!’ cried Sam.
‘What?’ shouted Tom Bombadil, leaping up in the air. ‘Old Man Willow? Naught worse than
that, eh? That can soon be mended. I know the tune for him. Old grey Willow-man! I’ll freeze his
marrow cold, if he don’t behave himself. I’ll sing his roots off. I’ll sing a wind up and blow leaf
and branch away. Old Man Willow!’ Setting down his lilies carefully on the grass, he ran to the
tree. There he saw Merry’s feet still sticking out – the rest had already been drawn further
inside. Tom put his mouth to the crack and began singing into it in a low voice. They could not
catch the words, but evidently Merry was aroused. His legs began to kick. Tom sprang away,
and breaking off a hanging branch smote the side of the willow with it. ‘You let them out again,
Old Man Willow!’ he said. ‘What be you a-thinking of? You should not be waking. Eat earth! Dig
deep! Drink water! Go to sleep! Bombadil is talking!’ He then seized Merry’s feet and drew him
out of the suddenly widening crack.
From” The Old Forest” chapter 1.06.