No, the weight has very little influence as long as the tyres are correctly inflated. In fact the Agattu is fitted as standard with very low rolling resistance Continental tyres which I praised in
my review for that very feature.
The only thing to avoid is tyres with poorly designed treads which have tread contact areas which constantly change during rotation, or tread which have discontinuities (gaps) for the bike to effectively drop into at each transfer from tread block to tread block. The knobbly mountain bike tyres often suffer from this poor design.
There are several elements to tyre rolling resistance:
1) The tyre is depressed at the point of road contact, and that uses energy in both deforming the tyre carcase and compressing the rubber content.
2) The tyre tread rubber compound has adhesion qualities to ensure it grips on the road, and that acts like glue, trying to hold the tyre in place on the road. A hard compound has the least "stiction" and poorest grip, a soft compound the most of both. Some two wheel tyres have dual tread compounds, hard at the centre for least rolling resistance when riding with the bike upright, softer at the slopes into the walls to give extra grip when banked on corners. Motorcycle race tyres sometimes have different soft compounds on the two tyre sides, depending on whether it's an anti clockwise or clockwise circuit being raced on.
3) Tyre treads can cause rolling resistance. Tread blocks with spacing are the worst, the wider the block spacing the worse they are. The first effect of these is that the bike falls off each block into the gap at each pass and then climbs onto the next block, so in effect the bike is climbing thousands of tiny hills all the time. Anyone who has cycled on the most knobbly mountain bike tyres will be familiar with this. The second effect of tread blocks is that the leading and trailing edges are deformed at each block mounting/dismounting, using energy to deform the rubber. Ribbed tyres are much more efficient, smooth slicks the most efficient, but treads can be efficient if the pattern of the blocks and spacings is angled and shaped to ensure a constant area of rubber is on the road all the time.
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for example)along the ROAD (as opposed to a track) using "Farmer giles off road knobblies" instead of the fitted thinish tyres. As the Aggatu(forinstance) is a little!!! heavier than a convential cycle, does the current tyre configuration have much impact on this 
