In the execution scene in Fantastic Beasts and Where to Find Them, when Newt takes down the guards, Tina is standing on the chair above the death potion and yelling for Newt to help her. Why couldn't she just Disapparate from it?
There are no canonical evidence on security setup at Magical Congress of the United States of America (MACUSA) headquarters. But based on the books there are places where Disapparation does not work.
Hogwarts does not allow Disapparation:
“He must have Disapparated, Severus, we should have left somebody in the room with him. When this gets out –"
"HE DIDN’T DISAPPARATE!" Snape roared, now very close at hand. "YOU CAN’T APPARATE OR DISAPPARATE INSIDE THIS CASTLE! THIS – HAS – SO”
Harry Potter And The Prisoner of Azkaban, Chapter 22, Owl Post Again
In fact, the Ministry of Magic had stopped Apparition in and out of the Ministry, after Voldemort had taken over. It is basically a security feature.
“We know everything important,” Harry went on, addressing Hermione. “We know they’ve stopped Apparition in and out of the Ministry; We know only the most senior Ministry members are allowed to connect their homes to the Floo Network now, because Ron heard those two Unspeakables complaining about it.“
Harry Potter and the Deathly Hallows, Chapter 12, Magic is Might
It's most probably because Gellert Grindelwald was trying to take over that MACUSA stopped Apparition.
She did not have her wand:
Wands make it easier to disapparate and Tina did not have her wand. Based on Does a wizard need a wand to Apparate?
The following day Snickerton led a band of men to arrest the accused witch. Cornered at her shop, she was unable to explain why she owned no scissors. Snickerton insisted that her hands should be tightly bound to stop her cursing them all and set about searching for the wand he swore he had seen her using. After several hours, however, he had found nothing suspicious, and his friends were starting to doubt him.
At last, Madam Crimp asked whether she might at least be unbound to scratch her chin, on which there was a large wart. Unable to see the harm in this, her foolish accusers agreed.
The moment Madam Crimp touched the wart, it sprang out of her face, revealing itself as her concealed wand. She promptly Disapparated along with a large bag of gold, and was never seen in London again.
Wonderbook: Book of Spells - Chapter 4
Disapparition from the death cell was almost certainly prevented.
The room Newt and Tina were in was a death cell, designed for the executions of wizard criminals. MACUSA would surely know that wizards capable of Apparating would attempt to Disapparate from the room and escape their impending death.
INT. CORRIDOR LEADING TO DEATH CELL—DAY
A long black metallic corridor leads into a pure white cell, which consists of a chair suspended magically over a square pool of rippling potion.
NEWT and TINA are forced into this room by the EXECUTIONERS. A guard stands at the door.”
- Fantastic Beasts and Where to Find Them: The Original Screenplay
Therefore, it would be a very egregious oversight to not prevent wizards from Disapparating from the death cell. It’s not impossibly difficult to prevent Apparition - most wizard houses do it.
“Professor, why couldn’t we just Apparate directly into your old colleague’s house?’
‘Because it would be quite as rude as kicking down the front door,’ said Dumbledore. ‘Courtesy dictates that we offer fellow wizards the opportunity of denying us entry. In any case, most wizarding dwellings are magically protected from unwanted Apparators. At Hogwarts, for instance –’
‘– you can’t Apparate anywhere inside the buildings or grounds,’ said Harry quickly. ‘Hermione Granger told me.”
- Harry Potter and the Half-Blood Prince, Chapter 4 (Horace Slughorn)
Considering that neither Apparition nor the means to prevent it are unobtainably difficult sorts of magic, MACUSA would almost certainly prevent wizards from escaping their death cell that way.