Have you checked for other threads in the forums? These threads may give you some additional hints..
Also a knowledge of which version of Server you're running would be useful.
FYI I have branched your query which you originally posted on Enabling subscription into this separate thread. By doing this, if there is something else that results in a correct answer to solve your problem you can mark it as such (a single thread can't have multiple correct answers).
Also by treating as a separate post, it isn't recognised as having been answered, so is more likely to attract responses from the community (it'll appear in the list of unanswered questions until its marked as correct/assumed answered).
Thanks for your help with this.
In regard to the Tableau version, both Desktop and Server are running on 10.0.2. Just a side note, we have not upgraded both Desktop and Server was due to the bug issue last year for 10.1.1, where the dashboards would not load on Tableau Server after publishing it via Desktop.
Would the Tableau version affect the functionality of the subscription icon?
Do you have your Configure settings for "SMTP Setup" set?
Also make sure you've allowed "Alerts and Subscription" by checking the boxes:
Go double-check your settings to make sure.
One way to verify if your SMTP settings are working is if you get the "Status change" email from your server(s) when ever they are started up: