Is it permissible to look at the deceased's belongings, such as reading their daily diaries?

It is not permissible to inspect what was a repository of the deceased's private matters and likely to contain something discrediting him, such as mobile phones, unless there is a legitimate Sharia (definition) reason, like ascertaining a right due to or from the deceased. For a Muslim retains sanctity after death. As for what did not contain private matters, there is no objection to examining it, and there is no problem in examining what the deceased endowed or published during his lifetime.