It depends on the rules of the grand jurisdiction. In Arkansas, we are required to conduct a criminal background check as part of the packet in submitting a petition for the symbolic degrees. (Shriners International requires the same). A completed packet indicating a past felony conviction is definitely a red flag. It's not an outright ban, but requires a lot harder look on the part of the investigating committee to determine the nature of the felony, how recent it was, and whether the petitioner has completed all parts of his sentence and probation, and had led a "clean" life since that time. In reality, it's usually cause for the committee to report "unfavorable" back to the lodge for balloting.
A brother who is arrested or indicted on a felony charge is, without exception, suspended by the grand master, and upon conviction, is expelled. Depending on the nature of the crime, the brother may be immediately expelled by the grand master. We had one of these events occur last fall when a wayward brother was arrested and jailed for capitol murder following a gunfight at a local gas station.