Performance issues after moving a database to SQL Server 2014 I've received a database that was previously on SQL Server 2008R2 but was just put on a SQL Server 2014 instance. There were no maintenance tasks run of any kind run on the database since 2014 (e.g. Rebuilding of indexes, updating statistics, etc.).

Once we ran update statistics as part of our regularly scheduled maintenance that we do on a set schedule, the performance of some queries has taken a massive hit to the point where some select statements will seem to never finish.

The queries have some CASE...WHEN statements in them, but I wouldn't expect there to be such a performance hit. Does anybody have any thoughts on what might cause such issues?

I've tried updating the compatibility level to 120 since it was on 100 when the database first came in but, that didn't make any difference on the performance.

Inculding links to executuion plans: 

Slow: https://www.brentozar.com/pastetheplan/?id=Byf1Manvg

Fast: https://www.brentozar.com/pastetheplan/?id=SkJB762De
Ok, at this point I would be considering "manually" replacing statistics to tide me over. I haven't done this myself, but it should work in theory...

You can confirm that this would work as follows:
1.  Restore the pre-upgrade database to your LIVE system and confirm that the query is fast in that database, and delete that database once done. (This test is to eliminate any additional environmental differences as the cause. If the query is slow, then the rest of this proposal may be useless.)
1a. Restore the pre-upgrade database to your test system and confirm that the query is fast. 
2. Update statistics, and confirm that the query is slow (cancel after a couple minutes of course)
3. Set compat to 120 and confirm that the query is still slow
4. Set compat back to 100 and confirm that the query is still slow
5. Restore another copy of the pre-upgrade database (I'll refer to this as Rest2, and the earlier as Rest1)
6. Extract all statistics from Rest2 using the techniques at https://thomaslarock.com/2015/07/how-to-recreate-sql-server-statistics-in-a-different-environment/ for all tables involved in the problematic queries (or all tables if that's simpler)
7. Apply the statistics to Rest1, and see if query is now fast (you may need to dbcc freeproccache first).

If it works, then it's almost certainly safe to apply the statistics to your live database--just make sure you have ONLY statistics scripted. And also set its compat level to 100. You should then see the queries run fast (though you may need to dbcc freeproccache first--but consider the possible effect on live performance).

Note I'm assuming here (based on your original post) that you do not have statistics autoupdate turned on, and that your data changes slowly enough that your old statistics will do until you figure out how to get your workload working with compat 120 and/or updated statistics (and you may as well sort both at this point).