For some reason a user may want to disable the RSS Feature in JIRA. It might happen by security concerns or product use guidelines. The RSS in JIRA can't be disabled on the Activity Stream, only for the issue navigator. There's already an improvement request for disable the RSS on ActivityStream, but it's not implemented yet. You can follow it here:

• JRA-33284 - Getting issue details... STATUS  

Resolution

• Browse to Administration -> Add-Ons -> Manage Add-Ons;
• Click in Show System Plugins;

• Use the search box filtering for Issue View Plugin;

• Click on "xx of 12 modules enabled" link;
• Now, look for "RSS (Issues)" and "RSS (Comments)". Click on the disable button for both of them.
  
  
  No need to restart JIRA.