Does Apple's TestFlight application auto update apps that external beta testers have installed?
They have to manually update it. A push notification and email is sent to let the user know that a new version is ready to be tested, but it's up to user to open test flight and install it.
As of November 11, 2020, TestFlight version 3.0 now supports automatic updates:
Source: https://www.macrumors.com/2020/11/11/testflight-app-automatic-update/