Do helicopters use more fuel when hovering?
Yes it is correct that helicopters use more fuel when hovering: the engine needs to apply more power to overcome drag. Here is a graph of the engine power required for different airspeeds, from J. Gordon Leishman, Principles Of Helicopter Aerodynamics:



The line for total power goes down between 0 - 70 kts with increasing airspeed, this is caused by the line for induced power: power required to overcome the induced drag of the helicopter blade. The total required engine power is the summation of:


Induced power. The power required to overcome the induced drag of lift creation, further detailed below. Propulsive power relates to the engine exhaust velocity now being helpful, and to the rise in induced power at higher speeds due to compressibility drag.
Profile power, required for the blade profile drag.
Parasitic power, for the drag caused by the air frame, rotor hub etc. Zero in the hover, very dominant at top speed. Helicopters have shapes that are way less aerodynamic than fixed wing planes, and this source of drag becomes very significant at higher speeds.
Tail rotor power. Up to 20% of the main rotor power both in the hover and at top speed, very low in the middle due to the helpful vertical tail. At top speed the torque of the main rotor is high and the tail rotor must do more work, unless the vertical tail can be adjusted.  




Induced power is dominant in the hover. Induced drag is caused by the backwards tilt of the lift vector: the higher the angle between blade and free stream, the more the vector is tilted backwards, which causes both loss of lift and increase of drag. The equation for lift L is:

$$ L = C_L \cdot \frac{1}{2} \cdot \rho \cdot V^2 \cdot S$$

and at a given altitude, the two variables here are $C_L$ (lift coefficient) and $V$ (airspeed at the blade). $C_L$ is an approximately linear function of angle of attack at the blade, so lift increases linearly with blade tilt-back and quadratically with increasing airspeed over the blade.



Above graph from Leishman shows the velocity distribution over the blades when hovering, and at airspeed. Quite a complicated situation - when hovering, the airspeed reaching the blade is only the rotational speed of the rotor, at forward speed the blade going forward has rotational speed plus airspeed. 

The helicopter does not roll over and both forward blade and retreating blade deliver the same amount of lift, with the rearward going blade tilted back more than it was in the hover. But the forward going blade is tilted back a lot less: airspeed has a quadratic influence.

Note that the circle in the plot at fwd airspeed is not stalled flow, but reverse flow: the air streams in at the back of the blade. So drag is now negative, the airstream helps to propel the blade! However there is loss of lift in the reverse flow area.

Induced power reduces with airspeed at first according to the simple 1-D impulse consideration (more air mass through the disk), and later increases as the disk is increasingly tilted forward and must do more work to overcome losses from rotor profile drag, airframe parasitic drag, and compressibility drag.

There is also an interference effect of the downwash over the fuselage: in the hover the air streams straight down, while in forward flight the rotor wash is more aligned with the fuselage, catching more of a streamline shape. Parasitic drag is of course dominant at top speed, while offloading the rotor by using fixed wing surfaces reduces the induced power at high speeds - but from hover to moderate forward speeds it is purely the reduction in lift induced power that creates translational lift.