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 effect is active while the Effect Monster Card is face-up on the field. The effect starts when the face-up monster appears on the field, and ends once that monster is gone or is no longer face-up; there is no trigger for its activation. These monsters are most useful if you have a strategy to protect them while they are on the field.
This is supported by previous rulings:
If "Banisher of the Radiance" and "Sangan" are destroyed simultaneously by an effect like "Lightning Vortex", then both are sent to the Graveyard and "Sangan's" effect activates normally.
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.
Destroy - A card is destroyed when it is sent to the Graveyard due to battle between monsters or by an effect that destroys a card. [...]
(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.)
Adding to user1873's answer, what you have to look at is the scope of the effect as well as the effect type of the Monster in question.
Unless an effect states so specifically or implies so, it does not work unless the Monster is face-up on the field. An example for stating so specifically is "When this Card leaves the field, ...". An example for implying so is "... Special Summon this Card ..." (a Card cannot be Special Summoned if it is on the field already).
In the specific case of Continous effects (like Banisher of the Radiance), these stop working as soon as the Card on the Field is destroyed (which happens just slightly before it's sent to the Graveyard from the rules' point of vew). This means that unless it implies so (Dark Magician of Chaos is an example of this, Battle Fader is another), a Monster's Continous effect that affects Cards leaving (or having left) the Field never applies to itself.
The example effect you provided would fall into the group "Trigger Effects". Those follow the same general rule, they have to specifically state so or imply so in order to work when the Monster is not face-up on the Field. Your example doesn't specifically state so and also doesn't imply so, which means it will stop working as soon as the Monster itself leaves the Field. This also means that it won't affect itself and only work on other Monsters which leave the Field while the Monster with this effect is still face-up on the Field.