You can burn subtitles with one of two filters: subtitles or ass.

subtitles filter

As of Nov 29, 2012 you can also simply use the subtitles filter:

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

This filter requires ffmpeg to be compiled with --enable-libass (if you are building FFmpeg yourself, look for the libass-dev package on Debian-like distributions).

ass filter

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

As with the subtitles filter, this filter requires ffmpeg to be compiled with --enable-libass.

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