It should be no. Sorry, I don't have the time to write anything detailed, so I'll just leave assembly instructions. Take a look at my answer here: equivalence of Grothendieck-style versus Cech-style sheaf cohomology describing an example due to Grothendieck where Čech differs from derived functor cohomology. If you write out what you get from the sequence $$0\to K\to \mathbb{Z}_X\to \mathbb{Z}_Y\to 0$$ described there, the Čech sequence should fail to be exact at $\check{H}^2(K)$.
Donu Arapura
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