How does cytochrome c oxidase inhibition cause cell death?
Inhibition is rarely binary - it's almost always subjected to stoichiometric effects. So the answer could be either option, depending on the amount of the inhibitor present in the system. If there is a huge amount of the inhibitor in the cell, it may more or less completely ablate ATP production by the ETC. 

On the other hand, if there isn't enough of the inhibitor to shut down all cytochrome C oxidase, you'd only see a reduction in the amount of ATP produced by the ETC (which could still be sufficient to kill the cell).

It also depends on the type of inhibition taking place. Is it reversible or does it bind permanently? Does it degrade quickly? These aspects are also important to consider.