The power required to maintain lift equal to weight¹ increases with climb rate and decreases with descent rate.
Thrust produced by a rotor (propeller) is equal to mass flow rate of air through the propeller disk times change in velocity of that air. This change in velocity increases momentum of the air (that balances the thrust), but it also increases its kinetic energy, and this energy has to be provided by the engine as induced power³.
But kinetic energy is equal to mass times square of velocity, which means the induced power is equal to mass flow rate times change in velocity times velocity² of the air. This velocity is some base value due to the rotor working, plus the vertical velocity of the craft.
So if the craft is climbing, the required power increases, and if the craft is descending, the power requirement decreases, even for the same lift.
If the angle of attack of the rotor blades is low enough so they don't stall when the flow direction changes to upward, the induced power will decrease below zero until it balances the parasite power³ and the rotor spins without any engine power at all. This is called autorotation, and allows helicopters to still make a safe landing in case of engine failure. However it requires the blades to have low enough angle of attack, so a quadcopter with fixed blade pitch won't autorotate, but the rotor will stall at higher rates of descent instead.
¹ Drag will increase the lift requirement in climb a bit, and decrease it in descent a bit, but with typical climb or descent rates – 100 ft/min ~ 1 knot – this is relatively small contribution.
² More precisely, $P = \dot m (v_0 \Delta v + \Delta v^2)$ where $P$ is the induced power, $\dot m$ is mass flow rate, $v_0$ is the air flow speed just above the rotor and $\Delta v$ is the increase in flow speed through the rotor. When the craft is descending, at some point $v_0$ will become negative, meaning the air flows from below.
³ Power required to turn a rotor (propeller) is induced power, which is the power required to increase kinetic energy of the air to produce thrust, plus parasite power, which is the power required to overcome parasite drag of the blades.