How authentic is the Hadith which states, in meaning, that the fire of the believer and the fire of the disbeliever shall not see each other?

The aforementioned hadith is weak because it is mursal (a hadith in which the Companion is missing from the chain of narration) and not musnad (a hadith with a complete chain of narration back to the Prophet).

As for the explanation of the hadith, there are two interpretations for the meaning of "their fires should not be visible to each other":

1. It is not permissible for a Muslim to reside in the lands of the polytheists where their fires can be seen by one another. "Seeing" here is a metaphor for close proximity. 2. It refers to the fire of war: Muslims invite to Allah, and polytheists invite to Satan, so they cannot agree. A Muslim should not reside among them. It is also said that this hadith was narrated concerning a people who embraced Islam in Mecca before the conquest.