  The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has
been utilized by many researchers to study important properties and invariants
of a manifold. Morse-Novikov cohomology is defined using the differential
$d_\omega=d+\omega\wedge$, where $\omega$ is a closed $1$-form. We study
Morse-Novikov cohomology relative to a foliation on a manifold and its homotopy
invariance and then extend it to more general type of forms on a Riemannian
foliation. We study the Laplacian and Hodge decompositions for the
corresponding differential operators on reduced leafwise Morse-Novikov
complexes. In the case of Riemannian foliations, we prove that the reduced
leafwise Morse-Novikov cohomology groups satisfy the Hodge theorem and
Poincar{\'e} duality. The resulting isomorphisms yield a Hodge diamond
structure for leafwise Morse-Novikov cohomology.
