Why don't the Iranians try to stop the airliner once it had taken off?
In real life the narrow escape at the airport didn't actually happen. In the context of the film, it can be reasoned that stopping the Swissair flight would have created a diplomatic incident and the Iranian revolutionaries would not have wished to anger Switzerland, which had much of Iran's oil wealth secreted in its banks by the Shah. It could also be construed that the guards at the airport might be reluctant to publicize to their superiors that they had allowed the Americans escape.
CIA operations specialist Tony Mendez (Ben Affleck) formulates a plan to enter Iran and rescue six Americans -- Joe (Scoot McNairy) and Kathy (Kerry Bishé) Stafford, Mark (Christopher Denham) and Cora (Clea DuVall) Lijek, Henry Lee Schatz (Rory Cochrane), and Bob Anders (Tate Donovan) -- who went into hiding following the storming of the U.S. Embassy in Tehran during the Iran hostage crisis (1979-1981). With the help of film producer Lester Siegel (Alan Arkin) and Hollywood make-up artist John Chambers (John Goodman), they concoct an elaborate ruse of a Canadian film crew scouting locations in Iran for a science fiction movie called Argo.
Did the British and New Zealand embassies really turn the fleeing American staff away?
No, the British Embassy actually sheltered them before moving them on to the Canadians where they could blend in better with their accents and which was not also under threat of being stormed in just the same manner as the American Embassy was. The New Zealand Embassy also gave help including smuggling the group to the airport for their escape.