How do I prove that the act of removing foreign keys doesn't corrupt existing data?
Implementing this stuff at an app level is a nightmare. You and your team will have to test, double check and retest code which does EXACTLY the same thing that's been done by MySQL (for InnoDB) for MILLIONS of users over a period of YEARS.

Follow the discussion (one of the best threads I've seen on stackoverflow) here. With all due respect to you and your team, does your boss REALLY think that you can write bug-free RI (Referential Integrity) code that has significant functionality in less than 5 years? I certainly don't! 

I can't tell you the number of times I've read on other forums (primarily Oracle) where some poor schmuck is crying his eyes out over an app he's inherited where RI was enforced at the app layer. Orphaned records, childless parents, inconsistent data...  the list goes on and on... data might as well be Swiss cheese, it's got that many holes! Tell your boss to have a read of this book by database professionals (Oracle experts, some of the best, most readable technical writing I've ever seen). From a review here - an outline of what Jonathan Lewis (a man who wrote a 530 page book on only (get this) the FUNDAMENTALS of the Oracle optimiser. 


  Chapter 10: Design Disasters, by Jonathan Lewis
  
  More war stories, for fans of Chapter 8! "Now prepare yourself to read
  all about 'The World's Worst Oracle Project.'" - Jonathan Lewis.
  
  This chapter describes some of the most common mistakes in development
  Oracle database applications. You'll certainly recognise some of them,
  because so many people stubbornly cling to certain beliefs. I know I
  like to bring up several of his points when I get into common
  arguments like these:
  1. We want our application to be "Database Independent."
  2. We will check data integrity at the application level instead of taking advantage of Oracle's constraint checking abilities.
  3. We want to use sequences for our primary keys.


Take CAREFUL note of point 2! If your boss persists with this madness, then my advice to you is to run fast and run far! You will spend your days in a miserable hellhole of constant firefighting, and never be able to fulfill your potential as a developer or a DBA and you'll learn very little! Just a few thoughts!