Using FFmpeg's ass video filter (see Wikipedia for  Advanced SubStation Alpha), we can draw text of the subtitles into the movie, like this:

ffmpeg -i video.avi -vf "ass=subtitle.ass" out.avi

Note: you need to have ffmpeg built with --enable-libass (if you are building FFmpeg yourself, look for the libass-dev package on debian-like distributions).

If your subtitle is in SubRip/MicroDVD or any other supported text subtitles, you have to convert it to ASS before using the filter:

ffmpeg -i subtitle.srt subtitle.ass

For windows users  this link may be useful, as you have to setup font paths just right to get libass to work.

With recent versions (starting Nov 29th 2012) of FFmpeg, you can also simply use the subtitles filter:

ffmpeg -i video.avi -vf subtitles=subtitle.srt out.avi