A number of paintings in this castle are believed to have been destroyed in a fire in 2009.

网友引用句子有二处错误，我已经改正。

这个句子相对应的主动语态句子为：

People believe that a number of paintings were destroyed in a fire in 2009.

改为被动句时有二种结构：

It is believed that a number of paintings were destroyed in a fire in 2009.

A number of paintings are believed to have been destroyed in a fire in 2009.

这种结构中使用不定式而不用现在分词，是believe这个动词句型要求的。类似的结构还有：

Sb/Sth is said/known/thought/reported to have doen/to be doing...

这种结构中都必须使用不定式，不能用现在分词。