I had this from 20 weeks pregnant until baby was 4 months old!
I tried stretches, heating pads, massages, and anything else I could think of. A lot of it had to do with hormones that eventually wore off (not much you can do about that), but I also worked on my posture which seemed to help almost immediately.
Nursing posture was the biggest issue. In those early months I was nursing 8 times a day for about 30 min. I sat cross legged on my bed most of the time (it was easiest for me back then). That meant I spent 4 hours every day hunched over with excess pressure on my tailbone. Not to mention all the time on the floor with the baby. I saw this image on how to sit so your pelvis is in a neutral position. I tried it out by sitting in a straight backed kitchen chair with my feet flat on the floor for an hour and the pain got considerably better right away. Over the next week or two I did my best to keep my back straight while sitting. Anytime my tailbone got sore I'd hand the baby to my husband and go sit at the kitchen table. I still nursed cross-legged, but I sat on the floor instead of on my bed so I could keep my spine aligned above my tailbone. Pretty soon the pain was gone and I stopped thinking about it.
Now I nurse pretty much anywhere and I don't send myself on "timeouts" to the kitchen table :)