That's not a thing. It's an urban legend.

Proof?

Pick up a bike wheel separated from a bike. (Or any wheel. Or fidget spinner. Anything that spins fast.)

Hold it still.

Now spin it really fast while still holding it. You'll see it didn't get any heavier.

Want a better test? Pick up a high RPM electric motor with a fairy heavy fly wheel. Something that spins REALLY fast. Hold it still, then turn it on. I guarantee you the spinning motor is no heavier.

Why? Because mass is a constant. Things don't get more massive, and thus don't get heavier, due to rotation or speed (at relativistic speeds. If you approach the speed of light, things change a little...but I ride a hybrid so I don't get going quite that fast)

EDIT: Fudge...I went to edit and error and deleted 80% of the post that cite all kinds of examples I don't feel like retyping.

Short version....

Rotational weight is technically a thing. But it's impact on the overall system of the bike is so ridiculously minuscule it's not even worth thinking about. And it ONLY applies to acceleration. (A heavier wheel it harder to get spinning) Once the wheel is spinning, its actually EASIER to KEEP it spinning if it's heavier at the outside of the circle. It's why fidget spinners have weights on the ends of the plastic arms. Take the weights out of a fidget spinner and it stops spinning much sooner.

Why? Because an object in motion stays in motion until acted upon by an opposing force. And the spinning circle that's heavier at its outer edge requires more opposing force to slow it down.