You can use a OnObjectEnter script trigger on the Date paid field to bypass the date paid field. This doesn't hide the field but bypass it if not paid.
If[Invoice::Paid="" or Paid="No"]
Go to next field
Also you can use a calculated value on the Date Paid field to auto enter the date.
Check the calculated value then enter the following :
If Paid is set to Yes then set to current date else it set the field to blank. Incase you select paid (Yes) then go back and change paid to No, it will blank the date.
There are a number of ways a layout object such as a field can be made to appear and disappear.
Here's a demo file that provides working examples of several such methods: https://www.dropbox.com/s/8levaz6deiyzjr8/LayoutObjectVisibilityControlDemo.fp7
If you are using FileMaker 12, Launch FileMaker and use Open from the File menu to open the file and get a copy converted to the FMP12 format.