Can a chicken's lard become liquid depending on its diet?
Without going into the biochemical details of fatty acid digestion, mobilization, and synthesis, the brief answer is that the animal regulates its own fatty acid composition to within a specified range regardless of its food. Lipid metabolism does not differ from the metabolism of other compounds, in that it is a highly regulated, enzyme-mediated process, which ultimately is genetically determined.

It must be stressed that a heterotroph assimilates its food, not as complete macromolecules, but as their base components. For example, the starch-fed chickens you described would digest the starch down completely to glucose, which would then be converted in the liver to fatty acids. These fatty acids would be, of course, those that the chicken had the enzymes to synthesize, which are consequently mostly saturated fatty acids of a certain length that happen to solidify when cooled.