If you drive the hearse it might. Seriously you should discuss this with someone who specializes in state licenses. A felony may or may not pose a problem; I simply don't know.
Like Mr. Kaman I honestly do not know what action if any the cemetery salesperson licensing board will react to a felony DUI conviction. Clearly if driving is part of the job it will make matters worse. I am not certain if there are any attorneys that would specialize in the cemetery salesperson board; however most licensing boards are similar and you just need an attorney that works with licensing boards.
My firm and I specialize in California occupational licensing law. To answer your question, criminal convictions that are "substantially related" to the qualifications, functions, or duties of a cemetery salesperson can trigger discipline. You are required to self-report all criminal convictions.
Generally, a DUI conviction will trigger a license investigation which can result in license restrictions and, depending on the circumstances, license revocation--felony convictions generally result in more serious licensing consequences than misdemeanors. So, find an experienced criminal defense attorney who understands licensing law or who is willing to work with a firm who does. Good representation can make all the difference in persuading a D.A. to wobble a felony DUI down to a misdemeanor. If you do incur a DUI conviction--felony or misdemeanor, all is not lost. Search "Blueprint for Success in licensing matters" for more information a help. Good luck!