Burnet, the town/county and the street, has always had the emphasis on the first syllable. The person for whom those named pronounced it that way. There are other people in history who had the same spelling who put the emphasis on the second syllable. Additionally, there are a couple other towns (Burkburnett, a tiny town on the Red River, for instance) that put the emphasis on "net" as well. It throws a lot of people off.
In addition to the streets, a lot of Central Texas towns are pronounced differently than the phonetic or linguistically accurate pronunciations. A few that come to mind:
Manor = "maine-er"
Salado = "sa-lay-do"
Llano = "lann-o"
Blanco = "blank-o"
Bourne = "burn-ee"
We have to leave a few off, of course, to give us more signs that you ain't from around here...