My grandmother, who grew up in western Pennsylvania, pronounced wash and Washington with an intrusive R: “warsh” and “Warshington.” Where does the intrusive R come from in that dialect? It doesn’t seem to be produced by the same mechanism that changes law and order to “lawr and order” in non-rhotic dialects (plus, my grandmother’s dialect was rhotic, if I recall correctly).
According to John Kelly of the Washington Post (Catching the Sounds of the City), he claims:
"warsh" is the predominant characteristic of what linguists call America's midland accent. The accent can be found in the swath of the country that extends west from Washington, taking in Maryland; southern Pennsylvania; West Virginia; parts of Virginia; southern Ohio, Indiana and Illinois; most of Missouri; and Kentucky, Tennessee, Arkansas, Oklahoma, much of Kansas and west Texas.
With the help of Barbara Johnstone, of Carnegie Mellon University, he traces it back to Scotch-Irish immigrants at least a couple hundred years ago.
Midland English is described as "firmly rhotic", where rhotic* (of or pertaining to a dialect of English in which the r is pronounced at the end of a syllable or before a consonant).
Barbara Johnstone, in an interview covered by the article Steel Speak said this:
But some features of the accent of southwestern Pennsylvania are geographically distributed in the same way—in the Pittsburgh area and to the west and the south— as are words and grammatical structures we know are Scotch-Irish in origin. This suggests that these may be older features that spread with the early settlers. One of these is the use of an r sound in the word wash, so that it sounds something like worsh.
(There's lots more linguistic trivia in that interview, and I think you'll find it interesting.)
*You may recall my mentioning rhoticity in reference to Why do British Singers Sound American? in response to this question. In that article, the author states that most dialects of England drop their Rs. Go figure, but that's not Scotch-Irish either.
Warsh was used frequently in Quyon and Mayo, West Quebec for at least two generations of Irish Canadians. My father and his siblings spoke what I though was simply bad English, however I now realize that it was an accent transported from Ireland (a la The Great Famine). They also said 'he's an arse', 'Warshington' and 'that'd be a fine how-do-you-do'. Clear, useful and to the point. I hope that dialect is still alive and kicking arse somewhere !