The phrase "tender is the night" comes from Keats's poem "Ode to a Nightingale." Several lines of the ode also form the epigraph to the novel, and many of its lines clearly apply to the atmosphere and characters, particularly the images of intoxication and mental breakdown.
The essential meaning of the phrase is clear enough: the night is tender, which is to say that it is both vulnerable and nurturing. The darkness of night hides a multitude of sins and wounds, and the characters in Fitzgerald's novel are fragile people, sorely in need of such concealment. This is particularly true of Nicole, who has been damaged by her father's abuse and suffers from neurosis.
Dick is also a vulnerable character and becomes more vulnerable over the course of the narrative. At first, he believes that he can care for Nicole and offer her some stability, but, in fact, he cannot even look after himself and becomes dependent on alcohol.
Superficially, these are glamorous, intriguing people, whose lives are enviable. It is through Fitzgerald's acute psychological examination that the reader comes to see how flawed and fragile they are. One might say that the author shines a merciless light upon his characters and that they cannot endure such exposure, since these are people who need the tenderness of night to preserve their illusions.