In a 747 is it normal for the Vr (takeoff speed) to vary greatly depending on fuel load and ambient temperature?
As Henning Makholm's answer discusses, the weight difference between the lowest and highest weights is significant. The operating empty weight (OEW) of a 747-400 is 394,100 lb, while the maximum takeoff weight (MTOW) is 875,000 lb. This means there is a difference of about 400,000 lb between the lightest and heaviest weights at which a 747 may be taking off, making the MTOW about twice as heavy as the OEW.

The rotation speed Vr is the point at which the aircraft can pitch up and develop enough lift to climb. So at MTOW, the 747 must develop about twice as much lift than closer to OEW. Most of this force is lift from the wings.

Lift can be calculated with the following equation:

L = 1/2ρv2sCL

L = lift
ρ = air density
v = airspeed
s = wing area
CL = lift coefficient

The air density will certainly play a role. Higher altitudes and temperatures reduce the amount of lift. But this example pertains to the other variables.

An airliner can use flaps and slats to increase the CL and s of its wings.

The CL will also depend on the angle of attack α of the wings. An approximate curve for the 747 can be seen here (it's for a 747-200, but it should be close enough). When an aircraft rotates, it is changing the AOA, which increases the lift. It is very important when designing an aircraft to make sure it can pitch up enough while on the runway to achieve the pitch angle needed for takeoff. A 747-400 rotates to about 10 degrees on takeoff. Based on the approximate CL-α diagram, the lift coefficient changes from about 0.3 to 1.25 when α changes from 0 to 10 on rotation.

When the aircraft weight is increased, flaps may be increased from 10 to 20. However, flaps 20 may be standard because these guys said so, so lets assume that CL and s will also not change. This leaves the airspeed v as the only parameter left to increase lift.

In this study, figure A-7 shows a distribution of ground speed at liftoff that lies between 140 and 190 kts. These numbers will probably be a little higher than Vr due to the acceleration between Vr and liftoff, and an average headwind. This means that at weights at the lowest end, 130 kts would be a reasonable estimate for Vr. 

So now lets make up some numbers. Lets take the weights 475,000 lb and 875,000 lb. Assume that the aircraft needs a lift 10% higher than its weight to take off. This gives us lift values of 522,500 lb and 962,500 lb. Using the lift equation and a low end speed of 130 kts to find the unknown portion.

522500 = 1/2ρ1302sCL

ρsCL = 61.8

Since we are assuming none of those vales will change, we can solve for v at the higher lift value.

962500 = 61.8/2v2

v = 176 kts

This is not too much lower than the high end liftoff value of 190 kts. So there you have it. Lift must be increased to account for the additional weight, and lift is proportional to the square of the airspeed.