Do SQL service packs effect SQL Compatibility mode? We will soon be upgrading from a SQL 2008 R2 instance to a SQL 2016 instance.

One of our databases has an application which our supplier says is compatible up to SQL 2012 SP1, but not SQL 2012 SP2, 2014 or 2016.

Once we upgrade to SQL 2016 and change the compatibility level of the database to 110 (2012), would the fact that the application is not compatible with SQL 2012 SP2 have any effect?
Most likely the application will work, however, generally when a vendor has this statement, they have not tested their application. They have not certified the application will still work once the SP or CU or version has been applied. Thusly, if you go to an "unsupported" version, they will not support you on any issues. 

I would not rely on a vendor's website documentation as the end all statement for supportability. Sometimes, they have not updated their documentation yet. Give them a call to determine what the true support options are, and then get that in writing.