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.