Contrary to upgrade a minor version (where the middle number changes) where no difficulty should be encountered, upgrade a major version (where the first number changes) is subject to significant modifications.
Update the libraries
First of all, you have to make sure you are using the last v2.x release.
You can adapt this command line depending on the other Flex servers you are using.
Upgrade the dependencies and the libraries
The version 3.0 now requires PHP 8.1. Please install this version or a newer one on your platform. Make sure the extensions are also installed. They are namely JSON and MBString. You may also need OpenSSL or Sodium depending on the algorithms you want to use.
If you require Symfony components, you shall have the version 5.4 or 6.x. for each of these components.
It is now time to upgrade the libraries. In your composer.json, change all web-token/* dependencies from v2.x to v3.0. When done, execute composer update.
You can also update all other dependencies if needed. You can list upgradable libraries by calling composer outdated. This step is not mandatory, but highly recommended.
Rector is a very nice tool from code upgrade. We highly recommend it.