Unquenchable fire is a biblical image of an awful and abiding judgment that cannot be extinguished. It describes the punishment and destruction associated with the final judgment of the wicked, without teaching annihilation in the supplied discussion.
John the Baptist uses the image for the burning of the chaff in Messianic judgment. Jesus applies it to Gehenna and speaks of fire that is not quenched; related New Testament passages associate fire with final punishment.
The expression is highly metaphorical and conveys the permanence and dread of judgment. The supplied material also notes Old Testament analogies involving fire that is not quenched.