InfluxDB returned error status: 401 Unauthorized

This is a known issue, and it has beed reported at grafana/grafana#37626, and it is due to the change of the model of managing the passwords internally by Grafana, according to grafana/grafana#37626 (comment).

How the error log looks like:

...
t=2021-11-11T08:00:20+0000 lvl=eror msg="Alert Rule Result Error" logger=alerting.evalContext ruleId=2 name="xxx: xxx" error="request handler error: failed to query data: InfluxDB returned error status: 401 Unauthorized" changing state to=alerting
...

Solution:

Go to your instance and login as administrator, then navigate to Configuration (engineering wheel) > Data Sources and re-type all passwords of your each of the datasources (specially those using the InfluxDB datasource).