Hello Again Folks,
I spent the day travelling to the Staunton Historical Foundation office, Staunton City Hall, and the Staunton Library. I found some interesting information, but not the "Dime Saloon" name I was hoping for.
First, I found out the current 111 S. Augusta Street was not the same 111 S. Augusta Street in 1881. The street addresses were renumbered around 1909. So the photo of the building we thought housed the saloon is not the same place. The site of the actual John Burns saloon is now a parking lot. The saloon was originally in the middle of a large brick brick building aptly named the "John Burns Building" that built in 1874 and demolished in 1972 to make room for a few extra parking spaces for city employees. The demolition of this historical building caused quite a stir and was the impetus behind the formation of the Staunton Historical Foundation.
Below is a portion of a Sanborn Insurance map from 1886 showing the location of the John Burns saloon (red arrow, 111 S. Augusta St). The current 111 S. Augusta Street building was not built yet (located near Lewis Creek in the lot marked 712).

Next is an image from a 1891 Sanborn map showing the location of the John Burns Saloon (red arrow) and a new saloon at the 712 S. Augusta Street address (blue arrow), which is now the current 111-112 S. Augusta building (confused yet?).

Next is a photo of the John Burns Building in the 1960's (right side of photo). The saloon entrance was towards the center of the building where the awning and restaurant sign is shown. The building was "Mary's Restaurant" in the 1960's.

Next is a current photo (taken today) of the parking lot where the John Burns Building once stood.

Despite my best efforts, I was unable to find anything that showed the John Burns saloon name. I looked at old city directories, newspaper advertisements, and recorded deeds. City Hall did not have liquor license records, so that wasn't an option. There was no will or marriage license for John Burns in the Staunton Clerk's Office.
I only had 2 old city directories to examine (1883 and 1904). John Burns was shown as the saloon owner in 1883 but was no longer the saloon owner in 1904. On the 1909 Sanborn map, the address was still a saloon, but it was a restaurant in 1914. Based on the lack of a will, no marriage record, and no business listing in 1904, I'm thinking that John Burns left Staunton sometime between 1883 and 1904.
Sorry this is such a long post. Very interesting to me, but it sheds little additional information on the "Dime Saloon" story, except for a correction on the building location.
Thanks for everyone's help!