What is required of a worshiper who interrupts his prayer (obligatory or Friday prayer) due to an emergency, such as leaving to extinguish a fire? Does he complete the remainder of his prayer or repeat it entirely?

Extinguishing fires is a righteous act that necessitates interrupting prayer. However, it invalidates the prayer, and one must resume it anew. As for Friday prayer, if you catch one rak'ah of it after extinguishing the fire, you have caught it, and you should make up what you missed. If you did not catch anything of it, then pray for yourselves if the conditions for validity are met; otherwise, pray Dhuhr.