Does Princess Leia ever learn that Darth Vader is her father?

Yes, Luke tells her in The Return of the Jedi, on a footbridge in the Ewoks village.

First, he tell her that Vader is his father:

LEIA
Luke, tell me. What's troubling you?

LUKE
Vader is here...now, on this moon.

LEIA (alarmed)
How do you know?

LUKE
I felt his presence. He's come for me. He can
feel when I'm near. That's why I have to go.
(facing her) As long as I stay, I'm
endangering the group and our mission here.
(beat) I have to face him.

Leia is distraught, confused.

LEIA
Why?

Luke moves close and his manner is gentle. And very calm.

LUKE
He's my father.

LEIA
Your father?

Immediately after, he tells her that she is his sister (and thus Vader's daughter).

LUKE
There's more. It won't be easy for you to
hear it, but you must. If I don't make it
back, you're the only hope for the Alliance.

Leia is very disturbed by this. She moves away, as if to deny it.

LEIA
Luke, don't talk that way. You have a power
I--I don't understand and could never have.

LUKE
You're wrong, Leia. You have that power too.
In time you'll learn to use it as I have. The
Force is strong in my family. My father has
it...I have it...and...my sister has it.

Leia stares into his eyes. What she sees there frightens her. But
she doesn't draw away. She begins to understand.

LUKE
Yes. It's you Leia.

LEIA
I know. Somehow...I've always known.