Give user rights to delete addresses:

1. Open the security group.
2. Highlight Records and click Options.
3. Highlight Constituent in the left frame.
4. In the right frame, mark View, Add, Edit and Delete for Address.

If the user has rights, check the indicator on the Addresses tab. If it is "Business", delete the address from the Business button:
1. Select the Bio1 tab.
2. Click the Business button.
3. Click the delete icon (red x).
4. Click Yes when prompted "Are you sure you want to delete this relationship?"