A maintenance cost is a usually-optional cost required to keep a card on the field. These costs are typically paid during each of the controller's Standby Phases or End Phases. If a player cannot or does not pay for the maintenance cost of a card, then that card cannot remain on the field, usually being destroyed. A monster destroyed as a result of not paying (or choosing to pay) a maintenance cost is not considered to have been destroyed by a card effect, and the action required to pay the cost is also not treated as a card effect.
A maintenance cost is signified in a card's text by the following format: During each of your (Phase to pay the cost), destroy (or any other action) this card unless you (maintenance cost action). Maintenance costs cannot be negated and they do not activate or start a Chain Link.
Even if a card like "Skill Drain" or "Jinzo" is on the field and negating the card's effects, you still need to pay its maintenance cost. Also, cards such as "Card Guard" and "Field Barrier" cannot protect these cards from destroying themselves due to not paying their maintenance cost.
Mandatory maintenance costs
"Imperial Order" (after its functional erratum), "Degenerate Circuit", "Destructive Draw", "Mind Protector", "Red Screen", "Number 41: Baguska", and many "Archfiend" monsters have mandatory maintenance costs. Unlike most maintenance costs, the player does not have the option of choosing not the pay the cost and letting the card be destroyed. However, "Pandemonium" can be used to avoid paying these costs for "Archfiend" monsters.
In cases like these, if a player can no longer pay their costs, then these cards are destroyed. In the OCG, if a player has exactly the amount of Life Points left to pay the cost, he/she must lose the Duel.
For "Destructive Draw" only, if its controller cannot pay its maintenance cost, instead that player's Life Points become 0.