I have a need to burn in a time code to a video and am wondering if this is something that ffmpeg is capable of?
Short answer, no.
Long answer, yes, but not without using a separate library to create the frames with the rendered time code on them, with transparency filling the rest of the frame, then using FFmpeg to overlay the frames on the existing video. Off the top of my head I don't know how to do this, but I'm sure if you're creative you can figure it out.
Edit: I've been working on this problem because it is an interesting question/project for me. I have come a little further in the solution by writing a Perl script that will generate a .srt file with the time code embedded in it for any given video file from which FFmpeg is configured to be able to read the metadata. It uses the Video::FFmpeg library to read the duration and saves a subtitle file as ${video}.srt. This will make it so it will render automatically in Mplayer if you insert the following lines in your ~/.mplayer/config:
# select subtitle files automatically in the current directory, all files
# matching the basename of the current playing file
sub-fuzziness=1
Still working on how to position and overlay the rendered subtitles on a video and re-encode in the same format. I'll update this post as I know more.