If you use the Facebook app that comes with your phone, it will use data.
There are 3 different ways to send pictures to FB via text messaging though, and it uses TEXT, not data. When you log on to Facebook on the computer, go to Settings (or Preferences) and then Mobile settings. Set your phone up to receive and send FB text messages. Then you will have 3 ways - I've successfully used all three.
I sent the first to 32665, and I put different text in the Subject and text fields of the message. Both showed up in the caption, Subject text as one line, Text as the second line.
I sent the second to email@example.com - same results.
I sent the third to my personalized firstname.lastname@example.org - same results. Your personalized email is found somewhere in the mobile setup screens.
On my FB page, they showed up as one post, 3 photos. (I sent them within seconds of each other, so I suppose that's why they all showed up together.)
It looks like they all work, so use whichever is easiest for you! I have all three listed under my Facebook contact, but it doesn't seem to matter which is used, they all produced the same results. And it's all text, no data.