Apple restricts what call data 3rd-party apps such as Truecaller can access to provide Caller ID. Because of these limitations, you must keep the app in the background open, and never “kill” the app (by double-tapping the home button and swiping up) in order for the Caller ID feature to work as intended.
The latest version of the Truecaller iOS app (v12.0 & above) auto-updates Spam & Business Caller ID data in the background. It does it when the conditions are great, like your phone plugged into the charger / phone being connected to WiFi. Keeping the Background App Refresh enabled makes sure that your Caller ID is always up to date.
Please make sure that Background App Refresh is enabled by going to iPhone Settings > General > Background App Refresh > Find Truecaller and turn the switch on.
It is recommended that you keep the app running in the background and never “kill” it (by swiping up) in order for the Caller ID feature to work better.