State Laws Differ
State laws govern probate courts, not federal law, so the rules can vary a great deal from jurisdiction to jurisdiction. For example, in New Jersey and Oregon, there are no laws stating that a convicted felon can't serve as executor of an estate. In New Jersey, this applies even if the named executor is still in jail when the testator dies. In other states, such as Illinois, the law specifically states that a convicted felon can't serve under any circumstances.
In states that allow the appointment of a felon, other probate laws might disqualify him from serving anyway. Many states require that executors post bond when assuming office, unless wills expressly indemnify them from having to do so. A bond is a sort of insurance policy against any wrongdoing by the executor and typically covers errors that might result in financial losses to the estate. If an insurance company refuses to issue such a bond because the executor is a convicted felon, the executor would be disqualified from serving unless the will waives the requirement.
In most states, it's possible for beneficiaries to object to the appointment of a will's executor by filing a petition with the court for his removal. However, they typically can't do so without a valid reason. They can't object because they don't like the person the testator chose, but they could do so if the nominated executor was convicted of a felony, especially if the conviction occurred between the time the testator wrote the will and the time of his death. Beneficiaries could argue the testator forgot to amend his will after the conviction and wouldn't have wanted the executor to serve under the circumstances.
In some states, the executor has a legal duty to come forward and tell the court about his felony conviction before taking office. If he doesn't and the court discovers or is made aware of the conviction, a judge can remove the executor from office even if he's already been sworn in. In this case, it might not be necessary for beneficiaries to file a petition for removal.
Courts may be more likely to remove an executor or bar him from taking office if the nature of his crime is specifically adverse to the welfare of the estate. For example, if the conviction was for embezzlement, this might create trust issues relating to the executor's involvement with the estate's finances. If the conviction was drug-related and had nothing to do with misappropriation of money, a court might be more likely to overlook it.