Keep the general implications in mind but, in particular, consider "Banisher of the Radiance". Its effect reads, "Any card sent to the Graveyard is removed from play instead." So does this monster go to the Graveyard or is it Removed from Play instead? As another example, suppose a monsters' effects read "If a monster on your side of the field is destroyed, you may pick a monster on the field and add DEF to it equal to the ATK of your destroyed monster." If this monster was destroyed, could you add its (non-zero) ATK to another monster's DEF?
Banisher of the Radiance wouldn't banish itself, it would go to the Graveyard. Banisher of the Radiance has a Continuous Effects, which is only active while it remains on the field face-up. (Rule book page 14).
This is supported by previous rulings:
Destroy means to send a card from the field to the Graveyard. Banisher of the Radiance prevents that from happening, so the hypothetical Card that you described would have no effect.
(Note: These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami. They were previously considered official and can still be correct, as long as they follow the official gameplay rules, or unless Konami has issued a ruling that says otherwise.)