This is covered by Rule 17 "Emergencies, Unfair Acts" in the 2022 NFL Rulebook; the crucial bits are:
ARTICLE 8. INTERRUPTED GAME. If, under emergency circumstances, an interrupted regular-season or post-season game
cannot be completed on the same day, such game will be rescheduled by the Commissioner and resumed at that point.
ARTICLE 11. GAME RESUMPTION. In all instances where a game is resumed after interruption, either on the same date or a
subsequent date, the resumption will begin at the point at which the game was interrupted. At the time of interruption, the Referee will call timeout and will make a record of the following: the team possessing the ball, direction in which its offense was headed, position of the ball on the field, down, distance, period, time remaining in the period, and any other pertinent information required for an efficient and equitable resumption of play.
i.e. the game will continue from the point at which it was suspended, with the Commissioner sorting out the details. Article 9 encourages the game to be played within 2 days at the original site if possible, but unsurprisingly gives the Commissioner discretion to do what is necessary.
In terms of previous incidents, NFL games have been suspended in the past due to the threat of lightning - for example, Titans at Dolphins in Week 1, 2018 - but I'm not aware of a game which was unable to be completed the same day.