The issue is most likely around the TraceFlag records that the developer console is creating and updating.
These can be accessed via the Tooling API, either the SOAP or the REST versions.
Probably the best starting point is to clear out any existing records. Especially those there the TracedEntityId and/or ScopeId matches the user Id in question.
Clearing the TraceFlags can be a bit tricky by hand. One way to do it is to use Workbench to invoke the Tooling REST API.
Once you have logged into work bench, use the REST Explorer to GET request to:
/services/data/v34.0/tooling/query/?q=SELECT%20Id%20FROM%20TraceFlag
Take the resulting ID's and send them in a DELETE request to:
/services/data/v34.0/tooling/sobjects/TraceFlag/{ID}
where you have replaced {ID}
with the TraceFlag Id to delete.
Alternatively, you can use the free FuseIT SFDC Explorer tool to query the Tooling API for the ID's. Then right click in the Id column and select Delete Entity
. (Disclosure: This is a product from my current employer)
Opening the Developer Console again should create new TraceFlags with an expiration date in the future.
If the TraceFlags don't fix the issue, you may be having issues with the Streaming API. It is used by the developer console to detect when new logs become available.
Try the following SOQL query when you expect to see a new log entry:
Select Id,LogUserId,LogLength,Request,Operation,Status,StartTime,Location from ApexLog
If it appears there then there is a problem with the Streaming API. Maybe check the browsers developer tools for error messages.