In Star Trek III, Saavik tells David that Spock is about to go through Pon Faar, and that he will need to mate to survive. Later on, she is shown performing some kind of ritual with her hands with Spock which seems to calm him down quite a bit, but then the scene ends.
At this point, did they have intimate relations? The film really wasn't clear on this, and I couldn't tell if we were supposed to infer that or not.
You are lead to infer that Spock and Saavik did indeed participate in the physical and mental aspects of the pon farr ritual. Spock experienced an accelerated version of pon farr due to the Genesis planet's influence in Star Trek III: The Search for Spock, as a young man. He was aided by fellow half-Vulcan, Saavik.
Later that year, Saavik was transferred to the Grissom, where she further studied the Genesis Planet along with Kirk's son, David Marcus. There, they found the body of Captain Spock, who was believed lost. When Saavik and David found him, Spock's regenerated body was physiologically that of a child, but he aged rapidly. Saavik helped Spock through the agonies of the pon farr. After she and her companions were captured by Klingons, Saavik was almost murdered by a d'k tahg-wielding Klingon officer, though she was saved by David Marcus, who was instead violently killed by the Klingon. Saavik returned to Vulcan with the Enterprise crew, to reintegrate Spock's katra into his body, in 2285. (Star Trek III: The Search for Spock) --Memory Alpha -> Saavik entry
I refer you to this well-crafted and informative video:
Yes, intimate relations were inferred to commence as part of the ritual of which the hand-touching was the beginning.
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Although the film is quite coy and cuts away from the actual naughtiness, we're left under no illusions that they had (ahem) physical relations, that being the only known way to cure a Vulcan's 'pon farr' urges.
Spock reached up and gently touched her cheek. His fingers followed the upward stroke of her eyebrow, then curved down to caress her temple, as Saavik met the gaze of his gold-flecked brown eyes.
[several hours later later]
Saavik smoothed Spock's tangled hair. The fever had broken, the compulsion had left him. He slept, and he would live. She wondered if she had done him a kindness by saving his life. He was still completely vulnerable to the convulsions of Genesis, which would continue to torture him.
Star Trek: The Search for Spock - Official Novelisation
and
He is more in control now... intense. His eyes turn and lock on hers. Hands touching, they move together until their faces are brushing... and their fingers embrace.
[later]
CUT TO:
INT. CAVE - GENESIS - CLOSE - THE YOUNG SPOCK:
Morning light is streaming in. His eyes are closed, he breathes calmly. A hand, Saavik's, ENTERS SHOT, to sooth his brow and smooth his matted hair.
During an interview, the writer of Star Trek III, stated that Saavik was in fact pregnant with Spocks child. This is why she stayed behind on Vulcan when Kirk and his crew returned to face charges in Star Trek 4. Star Trek II, III and IV were written as a trilogy. Which is why each movie picked up where the other left off. Unfortunately they decided to leave it at that, and not develop the part of the story where Saavik is pregnant with Spock's child, or even disclose this fact to the audience.
There are bonus Featurette videos included with the latest 4k release of the 1-6 movie series. On the Star Trek IV movie, a featurette called the "Three Movie Saga", has all of this information covered by interviews. I believe Harve Bennett confirmed this information in one of the statements he made in that featurette. Harve Bennett did the Story, and the screenplay of the movie.
