As I noted here, there's no entry protecting the likeness of the Angel Moroni in the US PTO's trademark register, nor does one show up in a search of the database of state trademarks.

There may be a common law trademark cause of action or a copyright claim, but there appear to be no registered trademarks.

My understanding is that it can be both. Copyright protects original works of authorship, and trademarks protect symbols which have acquired secondary meaning as the identification of a business or product. Without getting into whether the LDS Church is a business, nothing stops the statue from falling into both categories.

Link didn't work in the previous comment. Here it is: http://www.sarwark.org/writings/2007/03/of-angels-coffee-and-trademarks.html