Not finishing what you're served in Japan is, approximately, the same comment as a $0 tip is in America, except about food quality rather than service. Culturally, the act of not eating something prepared for you reads like "Your food is inedible and unfit for human consumption" rather than "I wasn't that hungry" or "This particular dish, well, I wasn't quite feeling it." This will generally not endear you to fancy chefs, although you might get a bit of leeway if you look obviously foreign.
(Japanese people get in trouble on the tip thing all the time in the US, so much so that my travel agent helpfully tried to instruct me on "peculiar customs of Americans" when selling me a ticket to my hometown.)
(Japanese people get in trouble on the tip thing all the time in the US, so much so that my travel agent helpfully tried to instruct me on "peculiar customs of Americans" when selling me a ticket to my hometown.)