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:
 http://mathoverflow.net/questions/4214/equivalence-of-grothendieck-style-versus-cech-style-sheaf-cohomology/18674#18674
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 $\check{H}^2(K)$.