Generally, No. This did not happen.

While classes did have separate sections and areas of the ship they were not supposed to go to there is no real evidence they were locked in or prohibited from escaping. Most of the scary iron gates seen on Titanic and as depicted in the movie were actually there for health and security reasons- third class was almost entirely immigrants who wouldn't be screened until they reached port and thus were considered a health risk. Additionally they had gates to keep passengers out food storage and other sensitive areas.

We have multiple accounts of third class passengers making there way through unlocked gates on their way out of the ship. There is exactly one account of a third class passenger being locked out of a gate on which that scene is exaggerated from:

Mr. BUCKLEY Yes, sir. They tried to keep us down at first on our steerage deck. They did not want us to go up to the first-class place at all.

Senator SMITH Who tried to do that?

Mr. BUCKLEY I can not say who they were. I think they were sailors.

Senator SMITH What happened then? Did the steerage passengers try to get out?

Mr. BUCKLEY Yes; they did. There was one steerage passenger there, and he was getting up the steps, and just as he was going in a little gate a fellow came along and chucked him down; threw him down into the steerage place. This fellow got excited, and he ran after him, and he could not find him. He got up over the little gate. He did not find him.

Senator SMITH What gate do you mean?

Mr. BUCKLEY A little gate just at the top of the stairs going up into the first-class deck.

Senator SMITH There was a gate between the steerage and the first-class deck?

Mr. BUCKLEY Yes. The first-class deck was higher up than the steerage deck, and there were some steps leading up to it; 9 or 10 steps, and a gate just at the top of the steps.

Senator SMITH Was the gate locked?

Mr. BUCKLEY It was not locked at the time we made the attempt to get up there, but the sailor, or whoever he was, locked it. So that this fellow that went up after him broke the lock on it, and he went after the fellow that threw him down. He said if he could get hold of him he would throw him into the ocean.

This sailor has never been identified. Furthermore, as he testifies- they simply broke the gate and continued on so it was not a massive hindrance.

He also testified that the third class passengers "...had as much chance [of survival] as the first and second class passengers."

The biggest hurdle for third class passengers wasn't locked gates but language barriers, poor information, and simply not understanding the layout of the ship.

Also, sidenote, the lowest change of survival for any gender/class on the Titanic was actually Second Class men- not Third Class men. Yes- statistically you had a better chance of survival in steerage then in second class (as a man).

source: I created /r/rms_titanic