Other mistake: When House is inserting foil pieces in the wall socket to create a short circuit in order to ignite a roll of toilet paper, he touches both pieces of foil simultaneously, which should have electrocuted him. (00:30:30)
Here Kitty - S5-E18
Continuity mistake: Series 5, Episode 18, "Here Kitty": When Judy (the patient) holds the tape out for House before her bronchial attack, the way she holds it changes instantly. (00:13:45)





Answer: If he were caught, he would still be forced to serve the jail sentence he was given. In addition, he may be charged with various kinds of fraud: insurance fraud if someone collects on a life insurance policy, identity fraud if he gets fake documents, tax fraud, etc.
LorgSkyegon