This issue has puzzled me for a while. I have searched "Subscription Button" on the web to no avail. Today's I searched "Subscription Icon" and I found the solution provided by John Liptak. The reason is that the server needs "at least one valid schedule for Subscription" for the button to show up. This helped me solving the problem! Later I found that Matt Coles gave a troubleshooting checklist which included the above as the last point. Most other suggested solutions or Tableau online help didn't mention the solution.

Just to document it here. The steps are (for server admins):

1.Create a schedule for Subscription on the Tableau Server

2.Make sure the above schedule is enabled. When a schedule is enabled, it has a Next Run Time. The Enable button in the menu is shaded.
If you disable the schedule, the next run time will not show. It's replaced by "Disabled".

Voila your subscription button will show up!
Just thought of an alternative design:
1.Always show the subscription button
2.When clicking on it, it shows "no schedule available, contact admin." if no schedule exist.

One of frequent tasks of managing a Tableau server is watching the failed data extracts and then fixing them. In a recent Tableau webinar, I presented a server admin view that helps monitor the data extract failure.