What happens?
Tennessee has a statute for that:
The effect of a sentence of imprisonment in the penitentiary is to put an end to the right of the inmate to execute the office of executor, administrator or guardian, fiduciary or conservator, and operates as a removal from office.
UPDATED TO ADD: The significance of this statute may not be immediately obvious. It creates a “particular civil disability that occurs upon the conviction and remains in effect throughout the defendant’s life unless restored by a specific statutory procedure.” State v. Johnson, 279 S.W.3d 522 (Tenn. 2002). Accordingly, the statute generally prohibits a person who has served time in prison from ever serving as an executor, administrator, guardian, conservator, or other fiduciary in the state of Tennessee.
Source: T.C.A. § 40-20-115