Since MongoDB 4.0.5 is officially supported for Sitecore 8.2.2, you should not encounter any issues during upgrade. Regarding MongoDB driver, you can find driver version compatibility version here. According to that table, the minimum driver version that supports both 3.6 and 4.0.5 is 2.7. However, MongoDB driver included in Sitecore rev. 161221 installation is 1.10, so most likely you will need to upgrade the dll.
Since Sitecore has implemented their own MongoDB provider, there is a good chance that you will not need to update MongoDB driver. If I were you, I would:
- Upgrade MongoDB to 4.0.5, but do not upgrade MongoDB.Driver.dll.
- Verify that Sitecore Analytics and session provider still works
- In case of any issues, upgrade the dll and contact Sitecore support since it would indicate that their documentation is not accurate.