Why did Leia not want to tell Han the truth about Luke and herself?

She is still processing the fact that her life has just been turned upside down and considering that both she, Gen. Solo and the rest of the gang are on the eve of a potentially war ending infiltration mission she is probably strong enough to compartmentalise all of this while the situation is still far from being concluded. To suddenly bombard Han with all of this (knowing that his first reaction may be to try and stop Luke from surrendering) is a distraction that they really do not need at this stage of the game.

This is a rather basic dramatic plot element: a trivial misunderstanding between lovers reveals their pride or strength of character and risks shattering their happiness. You will find it frequently, from kids' films like Shrek to romcoms to serious dramas. In Star Wars the misunderstanding is thankfully not milked for all it's worth. It's a Space Opera after all, not a contender for best drama.

It provides a way to build up tension between the two lovers and thereby creates suspense. It also creates a bond with the viewers since they are entrusted with privileged information. Viewers also witness a test of character and a bond deeper than mere sexual attraction: Han demonstrates that he is not a mere petty adventurer, that he has the confidence and required emotional sophistication to provide support and friendship when needed. It increases the viewers' affection for Han and respect for Leia's strength of character.

Any actual relevance to the logical development of the storyline is probably secondary. After all, if the Empire is nearing absolute control of the Universe, is this silly misunderstanding important?

In-universe answer:

There are many possible reasons why Leia might be reticent.

First, her feelings for Han may not have fully crystallized. Her relationship with Luke has been, well, very friendly. From Episode V:

INT. MILLENNIUM FALCON - SLEEPING QUARTERS

Luke rests on a cot, his injured arm wrapped in a protective cuff.
Leia gently wipes his face. The ship lurches again.

LEIA I'll be back.

She kisses him, then leaves the quarters.

But this is unlikely. Leia is a straight-shooter, and she has had a sisterly rapport with Luke (consider their first encounter on Episode IV).

So I favor a much better alternative. Being a Jedi is far from trivial. As Luke's experience shows, it requires more than fine-tuning telepathic skills to properly wield this power. And others can feel when you wield the force. Darth Vader and Luke have felt each others presence. Luke tells Leia of this:

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.

But Vader has not felt Leia's presence:

EMPEROR I told you to remain on the command ship.

VADER A small Rebel force has penetrated the shield and landed on Endor.

EMPEROR (no surprise) Yes, I know.

VADER (after a beat) My son is with them.

EMPEROR (very cool) Are you sure?

VADER I have felt him, my Master.

What's more, Luke explains to Leia her importance:

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 is unsure of the effect this information could have on her crew, including Han. It is no small feat to manage a rebellion being doggedly hunted down by the ruthless Empire. But there might be a better reason to keep quiet: she might become their last hope, and will have to keep hidden from the Empire while learning how to wield her Jedi powers.

Leia's identity is part of a lifelong secret, and she intuitively understands this

Luke (and the audience) know her identity is a great secret for her protection, as it is explained in the scene on Dagobah when Luke converses with the ghost of Kenobi.

BEN: Hmm. To protect you both from the Emperor, you were hidden from your father when you were born. The Emperor knew, as I did, if Anakin were to have any offspring, they would be a threat to him. That is the reason why your sister remains safely anonymous.

...

LUKE: But you can't let her get involved now, Ben. Vader will destroy her.

So Luke definitely knows to keep their relationship quiet. The question is-- how does Leia know? All he did was tell her that she was his sister; he did not tell her to keep it secret or why it would be necessary. So it is a good question.

The cynical explanation is that it is bad writing, and the screenwriter failed to differentiate between the audience's knowledge and Leia's.

A more charitable explanation can be discerned from Leia's immediate reaction to the astounding news:

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

She immediately understands that she and Luke are involved in an epic, lifelong secret-- a secret that would be between only the two of them. And she keeps that secret steadfastly, until the end of the film when the Emperor is dead and the danger is gone.

She does not yet know whether it is safe to reveal this secret.

All that she knows at that time is that the Death Star II and much of the imperial fleet has been destroyed. That in itself is much of a relief for the rebellion.

However, she cannot assume this means that both Vader and the Emperor have been killed. Those pesky Sith have a way of surviving! In ANH, Leia saw Vader aboard the first Death Star (when she was a prisoner there), yet Vader survived its destruction.

Furthermore, as loyal as Han's intentions may be, he is a poor choice to keep a secret from Vader. In ESB, Leia saw Vader torture Han to get at Luke. Should Vader survive this time, he might capture and torture Han again, to get to Luke, and then Leia's secret would be revealed. (Leia can resist interrogation by using the force, but Han cannot.) Better to not tell Han at all until Vader and the Emperor are destroyed.

Until someone (Luke) can confirm that Vader and the Emperor have died, it is not safe to tell anyone her secret.

Since you are asking about the movie version...

I have to say, while alsoa agreeing with other answers, that Leia was certainly also being mischievous and coy about the situation. IMHO the portrayal of Leia is very well done, she enjoys having the upper hand, playing both horses (both before and after she learned about her kinship with Luke). The moment you refer to, is a moment where Leia knows she can not keep the secret nor her true feelings hidden anymore.