Will SQL 2008 SSIS package files work with SQL 2016 databases after database upgrades?
At our location, we have SSIS running on SQL Server 2012. Our databases are on SQL Server 2016. We haven't had any issues with any of them, but then again, it really depends on what your packages are doing.

My recommendation is to have some sort of "sandbox" server and restore one of the databases and change the compatibility level to the latest level. Then test the SSIS packages to see if they would work.