A debate has raged for quite some time about whether or not Burke chairs were used on Star Trek: The Original Series. Most of the comments I've read in forums indicate a belief that they were, although at least one person claimed they were made by a company called Contemporary Shells.
The most compelling evidence for the Burke chairs having been the TOS chairs were concept sketches by Walter "Matt" Jeffries, the art director of the Star Trek series which I found at Star Trek Prop, Costume & Auction Authority.
I don't claim to know for sure. All I have are pictures of the Burke chairs and a couple of Star Trek photos, as well as the sketches by Jeffries, one of which is very clearly labeled "Modified Burke." I'll let you decide for yourself.
|Burke Model 115|
|One of our four Burke 116s|
(That's the previous owner in the background. This is his CL listing.)
|Jefffies concept sketch|
|Another Jeffries concept sketch|