We use access YouTube data via user, who authenticates the app. Replies, posted back to the channel are via this user profile.
It may be the user has no more rights to access YouTube content or user need to recall access to YouTube API.
1. To recall YouTube API Access user can follow the link https://myaccount.google.com/permissions and press REMOVE ACCESS
If you need to authenticate the YouTube access via different user, you can go to app settings Zendesk Admin Center -> Channel apps -> OAPPS Comments for YouTube -> Accounts tab, choose connected channel and press REAUTHORIZATION IN GOOGLE