I run Sublime Text 3 on a workstation that is very trimmed down, and I realize this is a strange request, but the method of updating to a new version appears to currently require a web browser to be installed.
This is the result of clicking the download button:
I think the download button should probably just do a wget (or similar) of the new version and then run dpkg on it, not present the user with a web page.
For reference, please see the efficacy of the "versionless" update system instituted in Chrome. Having silent, invisible updates ensures a much greater share of your users are running the latest version, and I think that would be an ideal feature to target for Sublime Text 3.
Customer support service by UserEcho