They can and will lose their tax exempt status or have it threatened if the government argues them to believe "extremist" for which false allegations would be an indication.
Losing the tax exempt status based on broad allegations have happened to the VVN-BdA the Association of people persecuted by the nazi regime, federation of antifascists. Based on it being mentioned in the "constitution protection report" of the interior intelligence in Bavaria the finance office of Berlin has revoked its tax exempt status in 2019. The organization was targeted as "leftist extremists" as they have a decisive anti-fascist stance, given that the organization has been founded by Holocaust survivors and their descendants are organized in it.
https://taz.de/Aberkennung-der-VVN-Gemeinnuetzigkeit/!5645383/