Do not disturb mode - but allow certain apps I want to turn on do not disturb mode on my iPhone. But we use a pager application for work, and I want that application to still be able to notify me and make noises.

How do I do that? I can only figure out how to allow contacts to not be in do not disturb mode.
Unfortunately, you cannot (on a non-jailbroken device).  There is currently no iPhone feature available to exempt a specific application from either Silent or Do Not Disturb mode.