I do have a common ground. I also tried running the remote control from the Arduino 5V pin instead of batteries - to make sure ground was definitely the same - and I still get the same results.
I did check my solder joints and couldn't see any problems. If it was shorted I think I would have had even more problems.
An optocoupler does sound like an interesting option - I might look into that. Although the original article did have problems with them.
Would a PNP transistor make a difference? Then I would holding it closed with a HIGH signal.
Thanks again!
Simon