User Defined Fields that are set to be required cannot have values removed when Mass Updating. You can remove that required setting by following these steps:
  1. Click Management
  2. Click User Defined Fields
  3. Click the Category the field is in
  4. Click the Name of the field
  5. Click Step 3 - Field Attributes in the Steps box
  6. Uncheck the box next to "Would you like this field to be required?"
  7. Click Save and Finish
Once you remove the required setting on the field, you can continue with your Mass Update to remove a value. After you remove that value, you can edit the User Defined Field again to turn on the required setting again.

Please note: If you choose to make that field required again there may be some fields that are now empty. If your Mass Update removed the only value in that field, the next time you go to the record the field applies to you will be required to fill in a value before saving the record.