Although the Har Habayis may have more kedusha than a shul, a shul is a place of worship. The prohibition not to destroy is specific to places of worship. Therefore, even though it is not prohibited m’d’Oraisa to destroy parts of the Har Habayis, it is prohibited m’d’Oraisa to destroy any part of a beis knesses.
Unfortunately, this topic has recently surfaced in the news. The Supreme Court in Israel has ruled that Beis Knesses Ayelet HaShachar in Givat Ze’ev must be razed to the ground. The court accepted the claim of a group of Arabs that the shul was built on their private land. Members of the shul have claimed that they have already paid Arabs for the land. In addition the shul has been operating for more than 20 years. However, the court has ruled in favor of the Arabs and ordered its destruction. The left-wing organization that is representing the Arabs in this case is Yesh Din.
As we have pointed out in the article, destroying a shul is definitely prohibited, either m’d’Oraisa or m’d’Rabanan. May we be zoche to the days when our righteous judges are returned to us.