I’ve heard about remote, seldom-visited “hollers” and settlements in West Virginia that were unknown to the outside world until the 1960s and the War on Poverty opened them up – roads were built, electricity service extended, and so on – but that’s really the opposite of the phenomenon described by the OP., where known towns were slowly forgotten.
Hmm. I think a town disappearing by accident would be nearly impossible, but could it be done deliberately?
If the town was at the edge of a national forest, then perhaps some sabotage at the county tax assessor’s office could make it appear as though the town’s area was part of the federal land (which would be untaxed). The trees would provide some cover from air surveillance.
For heat, the town could be fortuitously situated on some sort of natural gas vein, or a geothermal outlet. For food, they could forage in nearby caves for mushrooms, and fatten up a flock of domesticated bats. Vitamin C could come from wild onions and strawberries.
Eventually, of course, they would become Mole People.
(I hope M. Night Shyamalan isn’t reading this!)
It’s been awhile since I watched Children of the Corn, but didn’t the children murder every adult in town at once? That’s not nearly the same thing as a town being slowly forgotten. Even the smallest, most isolated town has contacts with the outside world. Somebody would sure as hell notice if the inhabitants of an entire town suddenly stopped paying their taxes, utility bills, answering their phones, etc. In addition to random passers by there’s the people delivering mail and goods to the local shops.
The lovely Times Beach, Missouri
In this vein, I propose that our next Middle Tennessee Dopefest be held in scenic metropolitan Cainsville. 
Cainsville Populated Place Profile / Wilson County, Tennessee Data
Heck, I can’t even find Woodbury! How will I know when I’m in Cainsville? Shall we meet at the hole in the road or the big burl?
nevermind
Out away from the bull in the pasture. 
In Oklahoma there used to be a town–a very small town–called Slapout. There were road signs telling you how far to Slapout (as if anyone cared). There was a store and a gas station and, I think, a post office.
It even used to have a website!
I just tried to find it on Google maps. It’s not there! (Too bad. Wouldn’t it be fun to have a Dopefest in Slapout? Well, okay…probably not.)
Another nominee: Homer. This one’s still on the map, but it was abandoned even in my youth. It was one of Oklahoma’s black towns, a result of segregation. Some of those traditionally black towns still exist, but not Homer.
Great idea, missred, and unless I’ve forgotten my Tennessee geography, that can’t be far from the dead center of the state. Is there a “far tar” nearby? If not, we need to have the Dopefest at the Cainsville Memorial “curve ahead” sign. 
There is (or was) a town by the name of Slapout in Alabama, too. Northeast of Montgomery in the general vicinity of somewhere between Wetumpka and Eclectic, IIRC. It was a well-known joke in those parts. But even though I have yet to find a map that shows it, this page at least makes reference to it and that it’s in the Wetumpka vicinity. It may well be one of those disappeared towns the OP is after.
There may even be an Alabama place called Plumnelly, though that could be fiction. Plum out of town and Nelly out of the world. Like the old thing about “not at the end of the world, but you can see the end from there.”
ETA: Also see Alabama Kitchen Sink: Slapout, Alabama
Some cities are under water as a result of dam building and flooding of once-occupied valleys.
California has a few of these.
With regard to the “Tax man would notice if the town stopped paying” theory - there’s a certain minimum cutoff on income tax. If the town had been essentially generating no real income for a while, there’s no reason they should even have been filing returns, and thusly, no reason for the IRS to notice they’d stopped.
While I can’t find Slapout, Okla. on Google maps, if you google “Slapout Okla” you get more than 6,000 hits, one of which says it’s the smallest town in Oklahoma.
Its web page might be one of those hits, but I didn’t go through all 6,000+ to find out.
It’s 1074 miles from Knockemstiff to Slapout.
Here’s the directions- Knockemstiff, OH to Slapout, OK
Its actual name is Holtville, but it’s far better known by its nickname.
You can even get a roadside view of Slapout, OK. It appears, however, that the gas station-grocery is out of business because there are no gas prices posted on the sign. The businesses across Hwy. 270 also look vacant.
My husband tells stories about people living in Coshocton County Ohio who were paid enormous (at the time) sums of money in exchange for walking away from their houses with only what they could gather up in five minutes. Mining companies were planning to strip mine in the area, and wanted everybody out. Apparently people did it - they say people walked out with half-eaten suppers in dishes still on the table.
I don’t know if these were rural houses or a small town, but that town certainly would have dropped away. Those houses would have been destroyed, though, so I’m not sure that’s what you’re looking for. (Never seen Children of the Corn. I don’t do horror movies. It’s not good for me.)