Jews do not destroy synagogues. Each shul is defined as a mikdash me'at, a miniature sanctuary and has intrinisic holiness. This issue arises in practice when a community is building a new shul to replace the old one. The old building may not be pulled down until the new one is completed (and the sanctity of the old is transferred to the new).
There are two reasons for this. First of all, the community might be tempted to divert the money raised for the new building to some other purpose and the replacement shul would never be built. Second, the community will be left without a shul to pray in. The exception, of course is, when the old building is structurally unsound. In that case you take it down and "work day and night to build the new one" (Talmd Bava Batra 3b).
When Israel withdrew from Gaza in 2006, some suggested it should destroy the 30 synagogues before leaving. In the end, the shuls were left intact because of the principle above. Hamas duly dynamited every one.