As discovered by developer Peter Steinberger, the message he encountered reads, “This app will not work with future versions of iOS. The developer of this app needs to update it to improve its compatibility.” While this seems to be more catered towards developers, apps that have yet to be updated will also be affected which in turn will result in iOS 10.3 “breaking” some apps, and in the process could inconvenience some users.
However to be fair, Apple has been rather patient on that front. The company first introduced 64-bit chipsets with the iPhone 5s back in 2013, so in reality developers have had 3 years to get their apps up to date and making them 64-bit. It is unclear as to how many apps out there have yet to be updated to 64-bit, but in 2015 Apple made it a requirement for all new apps to be 64-bit, and later extended that policy to cover app updates.
This means that there is a good chance that most apps are already 64-bit ready, so unless the app hasn’t been updated for years, you don’t have anything to worry about.