Should it not be "feed a cold TO starve a fever"?
Regarding the saying "feed a cold, starve a fever", as I understand, that saying has been dangerously abbreviated over time. The verbose version of the saying, that reflects the original meaning, might go something like "feed a [person suffering with a] cold in order to mitigate the fever".
So thank you so much for that, Douglas. That's not something we'd heard of. It sounds highly plausible that it should be feed a cold TO starve a fever because, as we have pointed out previously, you need to eat properly to mount an effective immune response, without which you might well develop a more severe fever.