For the tracking server? Yes, it does produce logs which you could see if running the tracking server as a standalone service. They are not exposed from the hosted tracking server in Databricks. However there typically aren't errors or logs of interest in the tracking server; errors of interest would occur (and be logged) from the MLflow client that is accessing it, for example, in a notebook cell.
For model serving, errors are likewise logged, but would be of interest and are exposed in the hosted MLflow serving service. You would see them in the Serving tab of a registered model. Or if deploying to Azure ML or SageMaker, these logs would likewise appear in the console of those services.
