Gemara

13a (Beraisa): After the Churban, if one was Makdish an animal, we lock it up so it will starve to death.

(Abaye): We do not cut it into two halves, for it says "You will break their altars... Lo Sa'asun Ken la'Shem."

45a (Mishnah): If Nochrim worship mountains and hills, they are permitted, but what is on them is Asur - "Lo Sachmod Kesef v'Zahav Aleihem";

R. Yosi ha'Galili says, "Eloheihem Al he'Harim" - their gods on the mountains, the mountains are not their gods.

Makos 22a (Rav Acha brei d'Rava): If one cooked with wood of Hekdesh, he is lashed for "va'Ashereihem Tisrefun ba'Esh (Lo Sa'asun Ken la'Shem Elokeichem)."

(R. Chananya): He can also transgress erasing Hash-m's name (by plowing through it) - "v'Ibadtem Es Shemam... Lo Sa'asun Ken la'Shem."

Rishonim

Rambam (Hilchos Beis ha'Bechirah 1:17): One who breaks off in a destructive way a stone from the Mizbe'ach, the entire Heichal, or (the floor) between the Ulam and Mizbe'ach or Azarah transgresses a Lav Lo Sa'asun Ken.

Radvaz: The Rambam did not list this among the Lavim in Hilchos Beis ha'Bechirah because it is included in the Lav to erase Hash-m's name, which he listed in Hilchos Yesodei ha'Torah. This is not Lav shebi'Chlalos (different Lavim learned from one verse; one is not lashed for such Lavim), for it is one Lav that forbids destroying anything on which Hash-m's name is called.

Kesef Mishneh: The Rambam forbids only in a destructive way. Surely, one may destroy in order to build.

Maharam Padavah (65): We can say that the Isur to break off from a Beis ha'Keneses applies only to something attached, just like the Isur to break from the Mizbe'ach, from which we learn.

Poskim

Rema (OC 152:1): One may not destroy anything from a Beis ha'Keneses unless it is in order to build.

Beis Yosef (151 DH Kasav ha'Mordechai): The Mordechai (826) says that a Beis ha'Keneses is called a small Mikdash, therefore, one may not break anything off a Beis ha'Keneses. The Sifri (Devarim 61) says that one who breaks off a stone from the Heichal, Mizbe'ach or Azarah transgresses Lo Sa'asun Ken. One may break off in order to build, for this is called building.

Taz (151:3): This is permitted when he builds and fills what he removed. If he breaks off to make a hole, it is forbidden, even if it is for a need. Therefore, one may not make a hole in the wall to hold up a board for a Stender.

Mishbetzos Zahav (3): One Lav forbids erasing Hash-m's name and Nesitzah (breaking off from the Mikdash). Just like the Isur to erase Hash-m's name is even in order to fix, the same applies to Nesitzah! It seems that one may destroy part of a Beis ha'Keneses in order to leave it that way, even if it is not destructive, such as for a Stender. This is unlike erasing Hash-m's name. That is forbidden even in order to fix. Surely one may break from a Beis ha'Keneses in order to fix it if it is in danger of falling.

Mishnah Berurah (12): Many Acharonim are lenient about making a hole to hold a Stender, unlike the Taz.

Kaf ha'Chayim (17): Some say that anything done for a need is constructive. It is like destroying in order to build. The custom is to break (holes in the wall) and make windows and extend ledges, and no one protests. One need not fill in the hole made.

Magen Avraham (152:6): The Isur to break off is only from something attached (Maharam Padavah 65).

Mishbetzos Zahav (ibid.): Maharam Padavah was unsure about this. This law is difficult, for one who burns Hekdesh wood is lashed, even though it is detached! The Isur is "you will burn their Asheros. Lo Sa'asun Ken." It is difficult to distinguish between an Asherah, which is normally burned detached, and breaking off rocks. Using Hekdesh wood to cook Chulin is not destructive, yet he is lashed. Perhaps this is even if he pays Hekdesh.

Bi'ur Halachah (152 DH v'Asur): Maharam Padavah never meant this. If so, one who breaks a Keli Shares in the Mikdash would not transgress this Lav! Surely this is wrong. Also, it says "destroy their gods... on the mountains", which refers to detached idolatry (Avodah Zarah 45a). The Torah concludes Lo Sa'asun Ken. Clearly, this applies to all Kedushah, attached or detached! Rather, regarding clearing benches and Kelim out of a Beis ha'Keneses, Maharam Padavah said that Nesitzah applies only to something attached. Since the Kelim are not attached, removing them does not harm the walls. There is no loss to the Kelim, so Nesitzah does not apply.

Mishnah Berurah (11): A Beis ha'Keneses or Beis Medrash is called a small Mikdash.

Bi'ur Halachah (DH Davar): Likewise, one may not destroy Kelim in a Beis ha'Keneses.

Bi'ur Halachah (DH Ela): The Shulchan Aruch permits to destroy an old Beis ha'Keneses after building a replacement. Even though nothing will be built after destroying it, this is considered 'in order to build', for they wanted to destroy the old one in order to build the new one, just Chachamim forbade to do so until the new one is built.

Kaf ha'Chayim (19): Preventing construction of a Beis ha'Keneses is like destroying it, but one is not lashed for this.

Kaf ha'Chayim (20): Eshel Avraham (6) was unsure whether one is lashed for breaking off from a Beis ha'Keneses. If he is, he is disqualified from testimony. Some say that the Isur is mid'Rabanan.

Question: A Tzibur rented a Beis ha'Keneses from a Nochri officer, and the rent became very high. They want to move the Kelim to elsewhere, for perhaps this will induce the landlord to accept less. Is this like destroying a Beis ha'Keneses before the new one is built?

Answer (Chasam Sofer OC 32): If the officer owns the building, this does not depend on whether rental acquires. Rental acquires like a temporary sale, but only for the term of the rental. Afterwards, it is the landlord's! It has no Kedushah, for one cannot forbid what does not belong to him. If a renter was Makdish, it is Kodesh as long as it is rented to him (Tosfos Erchin 21b DH Hachi ka'Omar). However, if the officer owned only the land, and the Yisre'elim built the building, it gets Kedushas Beis ha'Keneses. The concerns are 1) May one close the Beis ha'Keneses while trying to compromise with the officer? 2) Does removing Metaltelim from a Beis ha'Keneses look like destruction 3) Perhaps the officer will say 'take back your wood and stones', and the Beis ha'Keneses will be destroyed. Closing a Beis ha'Keneses is not destroying. As long as there is another place to pray, this is permitted. The Magen Avraham says that the Isur of destroying applies only to what is attached. Makos 22a connotes otherwise. Lo Sa'asun Ken forbids destroying Kedushah in a way parallel to burning Asheros, which are totally attached. We forbid even burning detached Hekdesh wood! All the more so, one who breaks off a Mizbe'ach, which was detached and then attached, transgresses. However, perhaps for detached Kedushah, one is liable only for burning, but not for breaking off or destroying. We learn from the fact that the Torah needed to forbid tearing the Me'il (Kohen Gadol's robe) or other Bigdei Kehunah. However, this is difficult, for the verse forbids burning and destroying together, without distinction. In truth, Maharam Padavah permitted removing Metaltelim (only) in a case when there is no damage to them. We may bring a proof from David ha'Melech, who took the Aron from Beis Shemesh to the house of Oved Edom, and from there to Ir David, before the Mikdash was built. Just like he was not concerned for uprooting Klei Kodesh from there place, it is permitted regarding a Beis ha'Keneses. If Yisre'elim would tell the officer to destroy the Beis ha'Keneses, this is telling a Nochri to transgress a Lav. This question was not settled; Poskim are stringent about it. Even if destroying a Beis ha'Keneses is mid'Rabanan, some forbid telling a Nochri to transgress a mid'Rabanan law even for a Mitzvah. If the Nochri will destroy it due to his anger at Yisrael, we are not concerned, even if we caused him to do so. Even the opinion that forbids causing a fire to extinguish on Shabbos permits causing Hash-m's name to be erased, for it says Lo Sa'asun Ken. Only an overt act is forbidden. The same applies to causing Nesitzah. Rashi (Devarim 12:4) brings a Sifri that expounds 'do not cause destruction of the Mikdash through Aveiros. This is Agadata, and we do not learn from it.