I am also very interested in "What is known about Chech cohomology of such a sheaf?", so if you got more information I would be interested.

What I can say is that Basic cohomology of foliated manifolds gives a resolution for that sheaf, and it is more related to what Chris said about $M/\mathcal{F}$. But I think that the resolution is only flabby (with no more hypothesis), not sure though.

  Basic cohomology can be infinite dimensional, so it can or can not satisfy a Poincarè duality. It seems to be well cover in the literature the riemannian foliation case.

Another resolution, which is fine and torsionless, is given by foliated ("tangential" sometimes is used to refer to it) cohomology.

Transversal structures gives a great deal of information as well.

I am also very interested in "What is known about Chech cohomology of such a sheaf?", so if you got more information I would be interested.

A fine and torsionless resolution is given by foliated ("tangential" sometimes is used to refer to it) cohomology.

Transversal structures gives a great deal of information as well.

More related to what Chris said about $M/\mathcal{F}$ is Basic cohomology of foliated manifolds. Basic cohomology can be infinite dimensional, so it can or can not satisfy a Poincarè duality. It seems to be well cover in the literature the riemannian foliation case.