There are no circumstances under which two identical bicycles (with differing rim weights) ridden by identical riders, using identical power, over an identical distance, will use a different amount of energy.
Take 2 identical bikes, each powered by an electric motor that delivers a constant power.
From the off, the bikes will accelerate until they reach a speed at which the constant input power is balanced by drag (aerodynamic, mostly). However the bike with the heavy rims will have taken fractionally longer to reach that constant speed, as it's had to supply extra energy to get the heavier rims rotating.*
Thus the bike with the light rims will reach the end slightly sooner than the bike with the heavy rims, leaving the bike with the heavy rims being powered for a little longer, using more energy.
* kinetic energy = 0.5 * m * v * v
Bike 1 Frame=11kg, tyres/rims=1kg
energy to accelerate to 10m/s (22mph) = 550 + 2*50 = 650J
Bike 2 Frame=10kg, tyres/rims=2kg
energy to accelerate to 10m/s = 500+2*100 = 700J
at 100W, that's an extra half second.
[edit]
I put "10 meters per second in miles per hour" into google, since the conversion factor isn't in my memory banks, and amongst the hits beneath the answer, the following Yahoo Answers page caught my eye, in which the "best answer" managed to calculate that half the speed of light was 26mph
http://answers.yahoo.com/question/index?qid=20130313231251AAPKwj3