There aren't really any full-time exorcists, and the profession doesn't exactly show up on Monster.com, but that doesn't mean there aren't people who see this career path as the one.
The majority of exorcists are part-timers, and spend the rest of their work hours doing the typical priestly duties of performing funerals, administering last rites, hearing confessions, etc.
The men of the cloth who do decide to undertake this extra-credit sort of mini-career can sometimes expect a certain amount of bonus pay, although that differs greatly from state-to-state, country-to-country, and even parish-to-parish. It also depends on the financial position of the one possessed and how badly they need their freedom.
That being said, consider this: a certain Reverend Bob Larson, who says he's performed more than 15,000 exorcisms (for an undisclosed price), charges $10 to take a questionnaire on his website to see if you might be possessed.
On his site you can buy any of his many books, purchase his two DVDs (as a boxed set), set up personal time with Reverend Bob (he charges airfare for cross-country exorcisms), and purchase a beautiful replica of Reverend Larson's Cross of Deliverance (a $100 "gift," shipping/handling not included). The good Reverend also holds more than sixty seminars each year.
So yes: in theory you can get rich being an exorcist, though you may want to skip that whole "poverty" part of the priestly oath.
Another thing to consider is the tax incentive: any sort of treatment that's required for on-the-job injuries (acidic vomit, spike impalement, loss of faith) might be—in-part or completely—tax deductible under the current tax code.