I know food possessiveness can be a problem and one show I was watching said that is a deciding factor in the dog being allowed to be adopted out, any food possessiveness and the dog will be PTS instead of adopted. This was the pounds policy in Detroit Michigan. I don't know if other places feel as strongly, but it did make me want to teach Joey not to be possessive with his food. There are steps you can take to teach them that you are the master and you are in charge of all food. One way is making them sit before you place the food bowl down, and not allowing them to take the food, until you give the release command. This sites has some other suggestions.
Learn How to Stop Dog Food Aggression or Possessiveness of Food like Cesar Millan Since your dog only does it with special treats, I wouldn’t worry too much, but sometimes you have t o take that special something away from them, and they should learn to give it up to you.