Just ran into the same problem today.
I know that the reason this used to work, up until now, is because of a third-party Quicktime extension called Perian. (Check your system preferences panel, do you see it there?) It wasn't a service being provided by Toast directly, it was Quicktime + Perian that was rendering the subtitles onto the picture, and Toast was just burning the result. (That's why there are no options in Toast to select or disable those subtitles; Toast doesn't know anything about them.)
Unfortunately Perian development was discontinued a few years back, but it's continued to work through 10.7 and 10.8. Perhaps 10.9 Mavericks has finally broken it? I'm not sure yet.
Either that, or recent versions of Toast are no longer using the same Quicktime pathway to render AVI movies?
Either way, I've got the same problem. Anybody have any suggestions?
There's a tool called Submerge (not free) that can hard-code subtitles onto a movie, but it doesn't output DVDs directly. You'd have to export an intermediate movie file with subs, and then burn that in Toast, as two steps (and two compression generations!). Not so cool...