Presidents in the modern age are increasingly required to look presidential. No baldies or beards, for example, since Eisenhower and some dude in the nineteenth century. Perry and Romney are almost made in a presidential Ken factory - and both presumably dye their hair (as, obviously, did Reagan). Looks, in other words, matter on an unconscious level in a president. We respond to these signals before our frontal cortexes kick in.John Sides reminds us, this is empirically wrong on the specific area of girth: overweight male candidates are evaluated somewhat more positively than thin ones (although overweight female candidates suffer a penalty). Besides, we've actually had overweight presidents. Taft was enormous (this was hardly a secret even before television), and Clinton struggled with his weight before heart troubles forced him onto a strict low-fat diet. (Does he look more presidential now that's he's gaunt?) So to say that overweight men don't look presidential seems a bit odd when we've had overweight presidents.
More generally, though, the idea that candidates have to "look presidential" is highly problematic, giving sanction to all sorts of bigotry. It's hard to separate "looking presidential" from "looking like the presidents we've already had," which leads to some uncomfortable areas. It wasn't too long ago that a sizable chunk of Americans wouldn't have found an African American "presidential" looking. No doubt many feel that way about women today. And Jews. And short people. And people with disabilities. Would that ultimately affect many people's votes? My guess is that it would be hard to find a measurable "presidential-looking" effect that moves votes beyond the major influences we can detect (economic growth/recession, war/peace, extremism/moderation). But maybe there is such an effect. Should we cater to it?
Do parties consciously nominate slim candidates, concerned that voters will be turned off by heavy ones? I really don't know. But if so, this sounds like bigotry, plain and simple. It's no different from Spencer Tracy in "Guess Who's Coming to Dinner" opposing his daughter's marriage to a black man, not because he had any problem with blacks, but because he was concerned that other people did. Any way you slice it, you're disqualifying someone based on physical appearance.