What is the part of a bottle called where the liquid comes out?
I think you are looking for rim
the outer, often curved or circular, edge of something.
Or more simply , mouth, as shown in the picture below.
Mouth:
c : the opening of a container, the mouth of a bottle. (M-W)
This is the mouth of the bottle.
3 : something that resembles a mouth especially in affording entrance or exit: such as
[...]
c. : the opening of a container • the mouth of a bottle
from Merriam-Webster