Can I hide or delete a tribute if a rude remark is made on my loved ones page?

You as the account owner have the ability to delete any comments or photos that are left by others. They will not be notified if you delete their comments and/or photos. You also can “lock” your page so that no one can leave tributes or photos on your loved one's page. This is often wise to do some time after the funeral when the majority of people have left their tributes. If someone wants to leave something after you have locked the page, you can easily unlock the page momentarily for them or upload the content yourself.

See this article to learn how to change your tributes settings.