Other mistake: Amber states that Chase has a cat, even though in several other episodes Chase says he doesn't have a cat and wouldn't let Cameron get one because he hates cats. (00:16:10)
Continuity mistake: During the scene in which House is offering up liquor-flavored ice cream he sets the containers in a row down the table 1-2-3-4-5. He then puts spoons in the three containers closet to Foreman, 1-2-3. The shot changes and before anyone touches the spoons they are now in containers 1, 2, and 4.
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