For a stock MGA the most common cause of this problem would be the cut-off contact in the voltage regulator (control box) getting stuck shut so that it does not break the contact when the ignition switch it turned off. If you remove the cover from the control box you may be able to break this contact free with a small screwdriver. If so, cleaning the contact set with fine grain emery paper may cure the problem. Adjusting the air gap for this contact may also be in order. Refer to the service manual for the procedure and proper spacing of the contacts.

Some people (for various reasons) may retrofit an alternator to an MGA in place of the original generator. In this case, a likely cause of electrical run-on can result from a shorting type failure of a diode in the alternator. Operating the alternator with one failed diode will generally not cause any harm to the vehicle, just limit the charging current to about half normal, and may or may not cause the ignition light on the dash to glow. If your alternator is a serviceable type, you may be able to replace the diode bridge, and the part should cost under $10. If yours is a non-serviceable type of alternator, you will need a new or rebuilt/exchange unit.