In general, yes. Continuous effects are only applied while the monster is on the field, an activated effects (Ignition, Trigger and Quick) can only activate when the monster is on the field.
If they activate in other zones (Honest, while in hand, Zeta Reticulant, while in the graveyard, DD Survivor, while out of play) then it will specifically say so on the card.
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As Destiny Hero - Plasma's equip effect is a "Once Per Turn" effect, yes it can, and the existing monster is destroyed.
If the monster has a continuous effect, this ceases to apply once the monster leaves the field. If Jinzo is destroyed, for example, traps can once again be activated.If a monster has an activated effect, then chaining a destruction effect to this activation will not prevent that effect from resolving. As an example, if Chaos Sorcerer activates its Ignition Effect, or Caius the Shadow Monarch's effect is triggered, destroying them before their effects resolve, will not stop the resolution happening.Flip Effect monsters:These are the monsters that say "FLIP:" followed by their effect. When a face-down Flip Effect monster is attacked, it is flipped face-up at the start of the Damage Step, damage is calculated, and the monster is considered destroyed or not.After all of that resolves, the Flip Effect monster's effect activates. After that event resolves the Flip Effect monster is destroyed and sent to the Graveyard (and the Damage Step ends).(Note: Flip Summon is different than Flip Effects or effects triggered when flipped face up.)After Damage Calculation effects:Whether these effects work or not depend entirely on the type of effect. Let's say Wall of Illusion attacks and destroys a monster. Wall's effect activates after damage calculation, but since the monster was considered destroyed it does not return to the hand."Destroyed as a result of battle" monsters:Monsters like Giant Rat or Masked Dragon effects still activate at the end of the Damage Step they were destroyed as a result of battle in."When" Effect monsters:Cards like Peten the Dark Clown have effects that activate "when" it is sent to the Graveyard. But only if this is the last thing to happen! If anything resolves immediately after it is sent to the Graveyard you "miss the timing".If Peten was tributed for a Tribute Summon, the last thing that happens is the monster being summoned so its effect doesn't activate.(Confusing? Yes, but these types of cards are few and far between.)The exceptions to the above will be any resolution that specifically needs the activated monster to remain on the field, ie, the resolution somehow refers to or affects it. Elemental Hero Prisma, for example, needs to remain on the field, for his name changing effect to apply to him.
In general, once a monster effect activates, the only thing that will stop it is negation. Destroying it, banishing it or setting it, will not stop the resolution. Nor will taking control of it, which does not switch control of the 'effect' that's already on the chain. Now there are some effects which quite obviously need the activating monster to remain face-up on the field, such as ones that boost that monster's stats. These might fail to resolve in part or in full, if the monster is no longer face-up. There are a slim handful of effects that do not affect the activating monster but nonetheless require it to be face-up on the field for its effect to resolve. But in 99.9% of cases, it is not required.
Red Eyes Darkness Metal Dragon ability allows the user to special summon a dragon from their hand or graveyard once per turn. This effect how ever restarts upon it's destruction and being replaced to the field or being removed from the field and replayed. In any case or condition; As long as Red eyes is removed from the field and placed back on the field, that his effect can be reactivated. Why? The ability of any creature of their "once per turn effect" remains against it self once it has used it's effect and remains on the field. Regardless of being flipped face down and being re flipped face up. We already know that it is the very same creature, the very same card, the very monster who has used that effect for that turn. That being noted: When a monster is how ever replayed, it is taken as if the creature was never played before, at least to a point of condition. Other words: Red eyes darkness metal dragon has not used his effect yet currently that turn for it is taken as a brand new "Red eyes Darkness metal dragon" being played. There for his effect can be reactivated. This is due to a simple Yu-Gi-Oh combo called "Reset." Reset is evolving monsters being removed from the field and being replayed. The same as a card called "Honest." Even if you used the effect of honest that turn and sent it to the graveyard. For what ever so reason if the same honest was sent back to your hand on that very turn and you could reuse the honest card for what ever reason. Then you can. The "Once per turn." Only becomes once depending on the condition> Like wise as stated: If the monster that has used it remains on the field with out ever being removed, then the effect can only be used once per turn. (Unless some card out there breaks that...but till then.) One last example I like to point out and a very special one I like to state: Dasher. Destiny Hero- Dasher ability when in the grave yard allows you to special summon a monster that you drew during your draw phrase after you reveal them that monster that can be special summon. If Dasher, the very dasher was sent out from the graveyard and placed back in. His effect would reset once more and allow you to use his effect once again. Why? Once more: The monster Dasher even though we both know he did his effect and even though he was re sent to the graveyard, he was the one that used it. He is now treated as a replayed creature or a monster that has not yet used his effect. Besides: A creature ability is granted the very turn they are played. It is an automatic rule. Any monster with an effect has their ability upon played. Thus once more: When a monster is replayed after being removed from the field, the once per turn ability is reset.
"Relinquished" + "Thousand-Eyes Idol" As long as this card remains face-up on the field, other monsters cannot change their battle positions or attack. Select 1 monster on your opponent's side of the field, and equip it to this card (this effect can only be used once per turn, and you can only equip 1 monster at a time to this card). The ATK and DEF of this card become the same amounts as the monster equipped to this card. If this card is destroyed as a result of battle, the equipped monster is destroyed instead.