When I archive and publish my app to the App Store, I have the following error:
WARNING ITMS-90339: "Deprecated Info.plist Key. The Info.plist contains a key 'UIApplicationExitsOnSuspend' in bundle ar [ar.app] that will soon be unsupported. Remove the key, rebuild your app and resubmit."
But I don't have that key.
You do have that key. It's labeled as "Application does not run in background". Remove that key from Info.plist.
Here's a little trick. Select the Info.plist. Then right-click on any key or value and select "Show Raw Keys/Values". Now you will see UIApplicationExitsOnSuspend.
As @maddy suggested You need to remove "Application does not run in background" key in info.plist.
If you want to make sure that your app runs in background, you can use UIBackgroundModes key in Info.plist, you can refer to below link for examples: https://stackoverflow.com/a/3413828/4525717, if this key is included, Apple would need valid justification for App store Approval process.
