Similarly the wheel is either spinning or it's not. If it took a constant input of energy to keep something spinning then the earth would have come to a stop billions of years ago.

The only time that weight on the tires/wheels makes more of a difference than other weight is during the brief periods when you accelerate to regain speed after braking. That can certainly be a factor in a close criterium race where you're braking into each turn and accelerating out of it but it plays very little role in more typical regular bike rides. When on an extended climb it's just total weight that matters and on long flat rides with little braking small weight changes become insignificant.

OTOH, heavy tires will frequently also have high rolling resistance due to their construction (thick tread sipes that lose energy by squirming and stiff sidewalls that also lose energy when compressed in the contact patch area). That's not due directly to the higher weight but there's a high correlation between heavier tires and lower performance characteristics. But you can also have two tires with the same weight but quite different levels of rolling resistance.