The Kasteleyn matrix (for counting perfect matchings) and the Lindström-Gessel-Viennot matrix (for counting families of nonintersecting lattice paths) are tightly related, as observed many times by Greg Kuperberg (see, for instance, his answer to this question and also this paper of his).

In particular, Kuperberg describes a "pivoting" operation which can be used to transform a Kasteleyn matrix into a related Lindström-Gessel-Viennot matrix while preserving the determinant.

The eigenvalues of a Kasteleyn matrix, at least, seem to be natural things to study. I have "always" wondered: How are the eigenvalues of these two matrices related?

The pivoting operation seems to do something complicated to the eigenvalues. Indeed, I suspect the relationship is complicated anyway, since the Kasteleyn matrix is much larger than the associated Lindström-Gessel-Viennot matrix.

However, the determinant of a matrix is the product of its eigenvalues... also each pivoting operation is subtracting a rank-one matrix from a maximal minor; there do exist theorems relating eigenvalues of a matrix to those of maximal minors, such as the Cauchy Interlacing theorem for symmetric matrices.