Currently there is no option in TeamPulse to disable this field.
Anyway you can modify a file to disable the dropdown. You will need to go and open this file - "C:\Program Files (x86)\Telerik\TeamPulse\wwwroot\Scripts\wirejs\timesheet\timesheetController.js".
The file is minified, so it may look a little bit ugly. You need to find - dataValueField:"id" and add ",enable:false" after that, it must look like this:
This will disable the dropdown. Please note that the dropdown cannot be removed because the information for the current user is needed. Also this will not completely fix your problem, because any user can make an api call and will receive the timesheet information for others.
Also one more thing. All users will need to clear their browser cache, so they can download the modified "timesheetController.js".
Also take in mind that if you upgrade TeamPulse you will have to make the same changes again.
I hope that the information is useful for you. If not feel free to contact us at any time.