It’s not true. First, the Ipod full of music stores the same amount of information than the empty one. Any memory is the same. It’s full because it is according to the standard used to store information, But the memory is there storing all zeros or whatever noise or old content it had before.
They probably were thinking about the battery. It is said that a loaded battery weights more than a depleted one. But even then, it’s not true. The battery circulates electrons and the current is the same going in than going out. Although it’s possible to create an unbalance, it’s only a couple electrons and it has nothing to do with the load condition. It happens anytime.
The worst part is that, even if it’s true, it would be for anything with a memory or a battery. It’s not a special power of the Ipod. The Ipod is just an overpriced gadget.