If you lookup Briar Texas in Wikipedia, it will tell you Briar is a "Census-designated place" which if you lookup is defined as a name of a location the Census Bureau uses to gather statistics for - but also they are usually "populated areas that generally include one officially designated but currently unincorporated community, for which the CDP is named". Such is the case with Briar.
I moved "here" in 2009 (it still isn't 100% clear to me if my property was ever officially in Briar city limits when Briar existed, if not I would have been right on the border). There is still a sign on FM 730N marking the location of Briar, and when we first moved to the area I assumed that the City still existed, but it was just North of my residence. It didn't take long to figure out Briar didn't really exist anymore at all, and people generally considered the unincorporated area where I lived to be "Briar".
I have tried to do research online about the City of Briar and came up with absolutely zero - not when it was founded, not when it was dissolved - nothing. That's part of the problem, with no official records I can find online, people's memory of where exactly it was at differs. I keep looking for written records (the newspaper archives online have never gone far back enough) - because the bits and pieces of the story I have heard about the town and it's demise are awesome.
The story goes something like this - there was lots of "good old boy" corruption in little Briar in the City government and police department. The long time mayor was named Raymond Riley. Good old Ray was Mayor of Briar, Chief of police of Briar, and a Lieutenant in the Briar Fire Department - one person described him to me as basically "Boss Hogg".
At a City Council meeting in 1982, one of the residents was complaining about the corruption in the City and the police department and how the people running the city were not doing a good job. The City Council told them "You can't fight City Hall!", to which they replied "but you can get rid of it!". And that's what they did - a petition was gathered and a taken before a judge who allowed an election to be held to dissolve the City.
On April 2, 1983, the City of Briar was disincorporated in a landslide election, with the final vote of 495-120.