A simple question I think?

Energy = force times distance. In this situation the same load is carried over the same distance... IF the only resistance is a constant friction then the energy and power required remain the same independent of belt speed. In practice some of the drag will be viscous... increasing with the square of speed. I would expect a small increase in power required, probably less than 10 %
Regards Jonathan
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And power is force times velocity, no? -- Ed Ruf Lifetime AMA# 344007 ( snipped-for-privacy@EdwardG.Ruf.com)

One way that I thought of that you can prove it mathematically is to look at the motor. Electric motors have a power curve. The power curve describes its output toque as a function of its RPM's. Depending on the type of motor (I assume that its speed is a function of voltage input) the power or energy required to achieve a desired RPM can be a function of voltage or current. If you look at the motor and the friction on the conveyor as a function of both load and speed, you should be able to easily show an increase in power consumption.
However, everything that I have just said could be moot if to increase the conveyor's speed you have only changed out the gearbox and the motor running the system had more than enough torque available to handle the system. In that case, the energy would remain the same, or else not increase significantly.