Are the children of refugees allowed to stay in Germany if they are born in the country?

No. Even if a child is born in Germany, it will not automatically obtain a residence permit.

If both parents have a temporary residence permit or are still in the asylum procedure, then the child, too, may reside in Germany. Their child’s case is considered under a separate asylum procedure. The parents or the competent foreigners authority must report the birth of the child to the Federal Office for Migration and Refugees (BAMF).

If one of the parents is a German citizen, then the newborn child, too, will be granted German citizenship. For the children of foreign citizens, different regulations apply. Only if the father or mother has legally resided in Germany for at least eight years when the child is born, and if he or she has a permanent right of residence, will the child automatically be granted German citizenship.

The Federal Office for Migration and Refugees can provide more information on this topic:

http://www.bamf.de/EN/Fluechtlingsschutz/FamilienasylFamiliennachzug/familienasyl-familiennachzug-node.html

Further information on family reunification:

https://rumoursaboutgermany.info/rumours/do-all-refugees-in-germany-have-the-right-to-be-reunited-with-their-family-members/