I'm from the edge of the U.S. South (Texas), and I've never heard "eat on" used in connection with actual eating, so this must be a Deep South locution, to judge from StoneyB's interesting answer.
'Eat on' as a slang idiom
On the other hand the idiomatic expression "What's eating on you/him/her?" (as a variant of "What's eating you/her/him?") is widespread in Texas and I believe that I've also heard it used in other parts of the United States where I've lived (Washington, D.C., New York City, and San Francisco Bay Area). But it's unclear to me whether it arose independently of the literal dialectal sense of "eating on" or is a unique offshoot of the Southern wording that caught on everywhere.
J.E. Lighter, Random House Historical Dictionary of American Slang (1994) has this relevant entry for the verb eat:
eat v. 1. to annoy, bother, or vex.—occ[asionally] constr[ued] with on. [First citation, followed by citations involving "eat on":] 1892 S[tephen] Crane, Maggie 36: "Well," he growled, "what's eatin' yehs?" ... 1918 Gibbons Thought Wouldn't Fight 99: What's eating on you, kid? ... 1937 Steinbeck Mice & Men 59: What's Eatin' on Curley? 1939 O'Brien One Way Ticket 8: What the hell's eatin' on you now, fella? ... 1961 Terry Old Liberty 31: I listened and tried not to let their talk eat on me too bad.
The 1918 instance of "eating on" is from Floyd Gibson, "And They Thought We Wouldn't Fight" (1918) where it appears in the context of a conversation in a military barrack room between the author and an American named Budd English, whom the author identifies as having a "quaint Southwestern wit":
"That's Kid Ferguson, the pug," English whispered to me, and then in louder tones, he enquired, "What's eating on you, kid?"
"Aw. this bunk in the paper," replied Ferguson. Then he glared at me and enquired, "Did you write this stuff?"
Early Google Books matches for 'eat on' used in a literal sense
A slightly earlier instance of "eat on" in its literal sense appears in M.A. Yothers, "Bud Weevils and Other Bud-eating Insects of Washington" (February 1916):
The writer collected many specimens of this weevil on one- and two-year-old apple trees at Brewster, Washington, in April and May, 1911 and 1912. They were also found abundant on the wild sunflower, Balsmorrhiza sagittata, which grows in many parts of the sagebrush region in great abundance. At Brewster, Washington, where so many of these weevils were found the sunflower was very plentiful just outside the young orchards. It is quite certain that Mylacus saccatus feeds at least to a considerable extent on this wild sunflower, for on some of the flower heads the flowers were largely eaten away. In several instances they were observed eating on the flowers. Although many of the flower heads were examined no eggs or larvae of the weevils were found on or in them.
In context, it seems clear that "eating on" is equivalent to "eating at" or simply "eating," since the weevils are not being described as consuming a meal procured elsewhere on the flowers. The Washington in question is Washington state in the U.S. Pacific Northwest, not Washington, D.C., in the Mid-Atlantic Region.
An intriguing very early instance of "eat on" appears in "Habits [of the Locust] at Night," in First Annual Report of the United States Entomological Commission for the Year 1877 Relating to the Rocky Mountain Locust (1878):
Alden, Freeborn County [Minnesota].—Remain on the ground unless disturbed. Eat on what they light on.—{[Stephen] Cross [& Albert Lea, of Freeborn County]
The telegraphic style of this note is a complication here, but "eat on" again seems to mean "eat at" or "eat." Minnesota in in the northern Midwest of the United States.
From "Mrs. Grundy on the Adulteration of Food," in the English periodical Punch, volume 28 (1855), we have this instance:
Your pickles and preserves yon stains bright green for to draw people's eyes on 'em,/ Whereby they verdigris contains, and them as eats on 'em they pisen 'em.
It isn't clear whether "eats on 'em" here is a dialectal form of "eats of them" or whether "eats on" is equivalent to "eats" alone.
And jumping back another couple of centuries, we have this instance from Ben Jonson, The Magnetick Lady; or Humours Reconciled (1632/1641):
Compass. Thou malicious knight,/ Envious sir Moth, that eats on that which feeds thee,/ And frets her goodness that sustains thy being;/ What company of mankind would own thy brotherhood,/ But as thou hast a title to her blood,/ Whom thy ill-nature hath chose out t'insult on,/ And vex thus, for an accident inher house,/ As if it were her crime! good innocent lady.
Here "eats on" appears to be a synonym of "feeds on" as biologists (and others) use that term today.
Overall, Google Books searches turned up very few matches for "eat on" in the literal sense of "eat." The fact that three of the four earliest matches involve the feeding habits of insects (moths, locusts, and weevils) suggests that "eating on" may have some sense of nibbling. But the sample size is vanishingly small.
