In short: Compulsory, no; advisable, yes.
Assuming your apps are already targeting a supported version of .NET (currently >= 4.5.2), then retargeting your apps to 4.8 is not compulsory. Particularly if you do not plan to take advantage of any new features. See .NET docs:
4.x versions of the .NET Framework can be used to run applications built for the .NET Framework 4.0 through that version. For example, .NET Framework 4.7 can be used to run applications built for the .NET Framework 4.0 through 4.7. The latest version (the .NET Framework 4.8) can be used to run applications built with all versions of the .NET Framework starting with 4.0.
As a minimum, you could run some regression tests against your app running on 4.8 before going live. How much testing it takes to gain confidence, and the chances of breaking changes, depends largely on what version you are currently targeting.
Ideally however, if you are in a position to retarget, build and deploy for 4.8, then a lot of unexpected behaviour can be caught even earlier by the compiler. Note that there are sometimes changes that require code changes and again, the likelihood depends very much on what you target at present.
To get an idea of what work you might need to do in your code:
Also worth noting that the likelihood of experiencing problems running an existing 4.x app on 4.8 without retargeting is lesser because:
the .NET Framework uses quirked behavior to mimic the older targeted version. The app runs on the newer version but acts as if it's running on the older version. Many of the compatibility issues between versions of the .NET Framework are mitigated through this quirking model.
EDIT: As of 26.6.2020 I am not aware of any plans to accommodate testing apps running in an App Service on 4.8 before the upgrade date, which means you will only be able to test on 4.8 locally.