It would depend on how the media plays the story—if they suppress it or persuade the low-information voters that it is no big deal, then the Democrats may not make a fuss. They would figure that any information gleaned would be used against Republicans. On the other hand, they might be worried about the White House being able to blackmail them so they would toe the Obama administration line.
If it comes out that Obama did bug the cloak room phones, then that would mean he was listening on Democrats as well as Republicans. They aren’t going to stand for it. That’s why if this congressman has evidence of the bugging then he needs to produce it.