Fasting is a recurring theme, with both Cathy and Heathcliff refusing to take food at various points.
In Cathy's case, her refusal to eat is almost a tool to get what she wants. It is more cry for attention than any real wish to do herself damage. She does relent and begins to eat again after a few days, although by then it is too late.
For Heathcliff, fasting is the means to his end.
Both Emily and Charlotte Brontë suffered from anorexia, using food restriction in strikingly similar ways to the heroines of their novels. An in-depth look at self-starvation in the works and lives of the Brontës can be found here.
It would have been the perfect refuge for Cathy and Heathcliff.