Well, for example, if one of the signals is "I have data for you", the downstream logic may attempt to interact with the powered-down logic to retrieve it.
But the text should also mention the power implications. CMOS gates whose inputs are in the forbidden zone may draw static current or even oscillate. Even if this causes no logical problem, if defeats the power saving objective. But an AND with one of its inputs low doesn't do anything unusual, regardless of what its other input is doing.
Edit in response to comment.
If the gates of the N-channel transistors in the powered-down block float to 0, none of those transistors are on, so they don't pull the outputs of the block to 0. And since the block is powered off, there's no power to drive them to 1, either. So, where the outputs float depends entirely on leakage currents. Those are unpredictable: time and temperature dependent. A femtoamp point defect that randomly activates and deactivates could cause a great deal of activity on a floating node, while having no significance at all for a properly driven one.