I believe Mandrill automatically handles bounced emails though, e.g. they stop emailing that address on your behalf:
After a bounce occurs for a recipient email address, Mandrill stops attempting delivery of the email. Further email to that address will be rejected temporarily.
Mandrill utilizes a set of heuristics to determine whether a bounce should be classified as a soft or hard bounce. Generally, permanent errors such as invalid mailboxes will get classified as hard bounces, while quota errors and other temporary problems are treated as soft bounces. Hard bounce rejections last longer than soft bounce rejections, and Mandrill doesn’t automatically convert soft bounces to hard bounces. If you’re seeing a lot of soft bounces to an address, it may be a sign that the address is unusable.
Ideally this information that the email is no longer valid would eventually percolate up to your app as well, of course..