Updating Shop Theme

If I update my shop's theme, do I need to perform any adjustments or reinstallation of the Auto tags app?

Auto Tags doesn't make any code modifications in the store theme. So you don't need to take any action when updating the theme. 

However, if you're using Auto tags to tag your products, you may want to check the tag format your new theme uses and make sure you adjust the workflows in the Auto tags app according to the new theme.