After looking at the type of cost from the advanced gameplay section, I would say it would be an activation cost. Opponent could play Magic Jammer, discard his card for activation, you would still have to return one face-up level 5 dragon or higher to your hand.
Costs for cards come in three varieties.
The first kind is activation costs. These are paid when you activate (play) a card. Activation costs cannot be negated by cards that negate the effect of a card.
The second kind is "maintenance" costs. "Maintenance" costs are costs that are paid in order to keep a card on the field. Examples of cards with "maintenance" costs include most of the Archfiend monsters from the Dark Crisis expansion, "Imperial Order", "Messenger of Peace", "Mirror Wall", "Regulation of Tribe", "Armor Exe", and more.
"Maintenance" costs cannot be negated by card effects that negate the effects of another card. They can only be negated by a card that specifically does so (such as "Pandemonium").
Example #1: "Imperial Order" will negate the effects of "Messenger of Peace", but will not negate the "maintenance" cost of 100 Life Points during your Standby Phase. If you do not pay the 100 Life Points, then "Messenger of Peace" is destroyed (you always have the choice whether to pay or not to pay).
Example #2: "Jinzo" will negate the effects of "Imperial Order", but will not negate the "maintenance" cost of 700 Life Points during your Standby Phase. If you do not pay the 700 Life Points, then "Imperial Order" is destroyed (you always have the choice whether to pay or not to pay).
Example #3: "Skill Drain" will negate the effects of "Armor Exe" (so with "Skill Drain" active, "Armor Exe" can attack the same turn it is Summoned), but will not negate the "maintenance" cost of removing 1 Spell Counter during each Standby Phase. If you do not remove 1 Spell Counter, then "Armor Exe" is destroyed (you always have the choice whether to pay or not to pay).
Example #4: "Skill Drain" will negate the effects of "Terrorking Archfiend" (so "Terrorking Archfiend" will not negate the effects of monsters it destroys in battle, and will not get to roll its die when targeted by an opponent's card effect), but will not negate the "maintenance" cost of paying 800 Life Points during your Standby Phase. If you cannot pay the 800 Life Points, then "Terrorking Archfiend" is destroyed (you must pay if possible, because "Terrorking Archfiend" says "this is not optional").
The third kind of cost is non-"maintenance" costs for cards that remain on the field. Examples include paying Life Points to attack with "Dark Elf", "Jirai Gumo", or Toon Monsters. These are not "maintenance" costs because they are not costs that are paid to keep the card on the field. These costs are negated when the card's effect is negated. So if "Skill Drain" is active, for example, you do not have to pay to attack with "Dark Elf", "Jirai Gumo", or a Toon Monster
