As time went on, people began to notice that Frodo also showed signs of good ‘preservation’: outwardly he retained the appearance of a robust and energetic hobbit just out of his tweens. ‘Some folk have all the luck,’ they said; but it was not until Frodo approached the usually more sober age of fifty that they began to think it queer.
They just took away all his good lines. Imagine movie Frodo laying down the law to Gollum:
In the last need, Sméagol, I should put on the Precious; and the Precious mastered you long ago. If I, wearing it, were to command you, you would obey, even if it were to leap from a precipice or to cast yourself into the fire. And such would be my command. So have a care, Sméagol!
I think if they had gone with the wise and courageous Frodo from the books then Elijah Wood would have been better at delivering this sort of line than Martin Freeman.
No, I don’t think Frodo ever does tell him to jump.
More relevant is that Gollum swore an oath by the Ring not to harm Frodo, and Frodo warns him thay the Ring will hold him to his word cruelly. Then when Gollum does betray Frodo, Bam! Straight in the lava.