How do I make Sublime Text 2/3 not open previous tabs when I launch it?

There is actually a better solution implemented in Sublime Text. Select Preferences-> Settings - User and set remember_open_files and hot_exit to false. The resulting file should look like this:

{
    "remember_open_files": false,
    "hot_exit": false
}

(All the possible settings can be seen in Settings - Default.)

Just restart Sublime Text afterwards (it may be required to restart the program two times, one time to apply the settings) and it should start up without any open files.

Note: I only was able to test this with Sublime Text 3, but I am pretty sure that this settings exist in version 2 as well.

Sublime Text 2

Check the content of the directory ~/.config/sublime-text-2/Settings for a file called Session.sublime_session

If those files are included in there, remove them

Sublime Text 3

ST3 stores the session files under:

~/.config/sublime-text-3/Local/Session.sublime_session

In Sublime Text 3, below worked for me.

$ rm ~/.config/sublime-text-3/Local/Auto\ Save\ Session.sublime_session