Here's how you do it.
On a Mac, pretty much any time you're typing text (including when you're composing a Tweet) you can hit ⌥⌘T to bring up the "Special Characters" window, and from here you can find (and use) pretty much any Unicode character out there. Try typing "zero width space" into the search field:
and you'll find the elusive character… by definition it's kinda invisible. But you can add it to your favorite Special Characters, insert it into what you're typing directly from here, or copy it to the clipboard.
If you want to make it super-easy to use a zero-width space you can create a global text shortcut in the "Language and Text" area of the Mac's System Preferences. Here below I'm setting things up so that every time I type "zws" a zero-width space is inserted: