The filmmakers do have a point that Faramir does make a quick decision to forget the Ring that might be hard to put over in a movie. On the other hand, they could have had a tense psychological scene in the cave with his men and flashbacks to his father and Boromir to do it..
I'm not satisfied with either of the two answers thus far, because they fail to reference exactly that -- a 5-minute long, tense psychological flashback involving Faramir and his brother Boromir.
As numerous people who were unsatisfied with the changes to Faramir's character have pointed out over the years that this cut scene (from the Extended Edition) is essential to understanding film-Faramir's character and motivations, and is the #1 scene that should have been left in TTT.
The key phrase explaining Faramir's motivation here (from the book) is when Faramir is ostensibly tempted by the Ring in the cave, where he says
"here in the wild I have you.. a chance for Faramir, captain of Gondor, to show his quality."
In the book this is an ironic statement, as he quickly discounts the Ring, asserting that he would not pick up the Ring if he found it on the highway, which always bugged me about Faramir in the book. (it seems unrealistic -- consider for starters whether it would be better to leave it on the highway for some Orc to find?) Faramir in the book is very much a Mary Sue, with some evidence that he is an author stand-in of sorts.
PJ (perhaps clumsily) wanted to be more realistic with Faramir's character, while also highlighting Gollum's "mistreatment" as a major turning point for Gollum's character (his near-repentance on the Stairs being sadly left out of the third film, which is a glaring omission). It's a lot less clumsy with the scene left in, and a lot more psychologically motivated.
In the movie, "show his quality" is an explicit reference to the deleted scene, which explains Boromir's, Faramir's and Denethor's motivations.1 Here's an excerpt of the scene in full. I trust it clears up a great many things about these three characters in the film:
Two Towers Extended Edition Scene 41. "Sons of the Steward"
Faramir sees a vision of himself standing by a river bank as the body of Boromir floats past in the elvish boat. He looks shocked and grieved.
FARAMIR: His horn washed up upon the riverbank, about six days past. It was cloven in two. But more than this, I know it in my heart. (pauses) He was my brother.
INT. Henneth Annûn. Scene 41.
Faramir sits alone holding Boromir's cloven horn. He sees a flashback to when Osgiliath was retaken. Boromir stands up high on a battlement, and raises the banner of the Stewards of Gondor. The soliders cheer.
SOLDIERS: Boromir! Boromir!
Boromir puts the flag down and unsheathes his sword.
BOROMIR: This city was once the jewel of our kingdom. A place of light and beauty and music. And so it shall be once more! (he raises his sword and the crowds cheer) Let the armies of Mordor know this: Never again will the land of my people fall into enemy hands. (more cheers) This city of Osgiliath has been reclaimed... (he raises his sword again) for Gondor!
CROWD: For Gondor!
The crowd cheers again. Faramir appears and walks joyfully to Boromir. They hug, laughing
FARAMIR: Good speech. Nice and short.
BOROMIR: Leaves more time for drinking! (they laugh) Break out the ale! These men are thirsty!
The crowd cheers and Boromir brings two goblets of ale, giving one to Faramir.
BOROMIR: Remember today, little brother. Today, life is good.
They toast and take a drink. Faramir glances warily sideways --
BOROMIR: (laughing) What? FARAMIR: -- He's here.
BOROMIR: (seeing Denethor talking to the soldiers behind them) One moment of peace! can he not give us that?
DENETHOR: Where is he? Where is Gondor's finest? Where's my first-born?
BOROMIR: (looks jaded and turns to face his father) Father!
DENETHOR: They say you vanquished the enemy almost single-handedly.
BOROMIR: They exaggerate. The victory belongs to Faramir also.
He looks towards Faramir who walks towards them
DENETHOR: But for Faramir, this city would still be standing.
Faramir looks uneasy
DENETHOR: Were you not entrusted to protect it?
FARAMIR: I would have done, but our numbers were too few.
DENETHOR: Oh, too few. You let the enemy walk in and take it on a whim. Always you cast a poor reflection on me. (he walks up to Faramir. Boromir drops his head)
FARAMIR: That is not my intent.
BOROMIR: You give him no credit, and yet he tries to do your will.
Boromir walks away and Denethor follows him. BOROMIR: He loves you, Father.
DENETHOR: Do not trouble me with Faramir......I know his uses, and they are few. We have more urgent things to speak of. Elrond of Rivendell has called a meeting. He will not say why, but I have guessed its purpose. It is rumored that the weapon of the enemy has been found!
BOROMIR: (looks worried) The One Ring. lsildur's Bane.
DENETHOR: (whispers) lt has fallen into the hands of the Elves. Everyone will try to claim it: Men, Dwarves, wizards..! We cannot let that happen! This thing must come to Gondor.
BOROMIR: (shakes his head) Gondor.
DENETHOR: (he clutches Boromir's arm) It's dangerous, I know. Ever the Ring will seek to corrupt the hearts of lesser Men. But you, you are strong!
And our need is great. It is our blood which is being spilled, our people who are dying. Sauron is biding his time. He's massing fresh armies. He will return. And when he does, we will be powerless to stop him! You must go. (Boromir looks taken aback) Bring me back this mighty gift!
BOROMIR: (shakes his head) No. My place is here with my people! (he walks away) Not in Rivendell.
DENETHOR: Would you deny your own father?
FARAMIR: If there is need to go to Rivendell......send me in his stead.
DENETHOR: You? Oh, I see. A chance for Faramir, captain of Gondor, to show his quality. [beat] I think not. (Faramir lowers his eyes, deeply hurt) I trust this mission only to your brother. The one who will not fail me.
Boromir is on his horse. He looks up at the Gondor banner, sadly, and then at his brother.
BOROMIR: Remember today, little brother. (He rides out of Osgiliath for Rivendell)
[This is the last time we see him in the series, and the last time Faramir sees him alive.]
Here's shots from the scene in context.
1 Note that in the film-verse, Elrond explicitly summons one of Denethor's sons to the council (not just through a dream) and Denethor already suspects that the Ring has been found. He has already instructed his elder son to bring it back to Gondor, and doesn't consider Faramir, the "wizard's pupil", up to the task.
So Faramir's motivations in the film are all about family dynamics.