Are restaurant health inspections necessary?

No, in fact they are harmful because they are ineffectual, mislead consumers, and are expensive to administrate.

Consider that if inspections were useful, then you would feel free to eat any restaurant you choose, without being concerned about the filthy-looking ones that passed their inspection anyway. As illustrated in this article from the NY Post:

February 25, 2007 -- A city health inspector gave a passing grade to the notoriously filthy, vermin-infested KFC/Taco Bell just one day before shuttering it - after news cameras recorded a rat rampage through the Greenwich Village restaurant.

"We're looking to see if the inspector dropped the ball on this," said Health Department spokesman Geoffrey Cowley. "I think it may not have been as rigorous an inspection as it should have been."

Restaurant cleanliness can be predicted from many things that anyone can observe, directly and indirectly: Garbage on the tables, roaches on the walls, odors, reputation of the franchise, attitude of the employees, visible appearance of the food, etc. And after accounting for those variables, how much extra could possibly be learned from a restaurant inspection by an unmotivated civil servant? Would you really trust the inspector's judgment over your own?