The basic idea of no-cloning is that, under unitary evolution you can't copy an unknown state. I don't precisely know what entropy related no-cloning theorem you are referring to. However, one uses no-cloning arguments for quantum capacity problems, which involve entropy terms.
Here is an example of that thinking: The capacity of a qauntum channel is characterized by an entropy difference (for basic definitions see 'Quantum Channel Capacities' by Graeme Smith (2010)). Using a no-cloning 'type' argument, for some channels this capacity must be zero, which gives a bound on the entropy difference. One place this no-cloning type argument is invoked is 'Capacities of Quantum Erasure Channels' by Bennett et al (1997). Here is how that works:
A quantum channel, has two outlets the output and the environment. Say, a channel for any input, sends exactly the same state to the output and the environment (this is possible). Then such a channel cannot have any quantum capacity. If it did, then over many uses of the channel, some set of unknown quantum states can be sent perfectly from input to output (by definition of quantum capacity). The same state can also be received at the environment (since output and environment receive exactly the same state). This would mean, we have copied an unknown input into two. This violates our regular no cloning, since the whole system (input = [output + environment]) evolves unitarily.