If anyone wants to chop up their clips (roughly) without having to re-encode them you could try using MP4Box GUI - http://www.videohelp.com/tools/My-MP4Box-GUI
Its nothing fancy but the split function allows you to extract between 2 timecodes without losing any quality.
I work with MKV's not MP4's so this is untested, but in theory will work.
Use mp4box and its GUI if you don't like the command line, to extract the subtitle you want to burn in.
Then use Subtitle Edit to open and save the subtitle as a SubStationAlpha (SSA) format.
Use mp4box again to put the SSA file into the MP4 file. Open it back up in Handbrake and the option to Burn In should now be available for that subtitle track (not the original, if you left it in).
Edit: Just remembered you said you had an AVI and SRT originally, use that original subtitle and save it as SSA. Not sure how to add an SSA to an AVI though, might still have to convert to mp4 first and then add it. Handbrake will not Burn In an SSA that is not in the file that you are converting. You can import SRT's but not SSA's, IIRC.
Ah, I see when I did it it was a different format.
Try this program. (It was recommended on a random forum.) http://www.videohelp.com/tools/XviD4PSP
If that fails.
Apparently, subtitle factory can convert SRT to SSA.
This can mux the SSA into the video. (It was recommended on a random forum.) http://www.videohelp.com/tools/My-MP4Box-GUI
Then you can hardcode it with Handbrake.
You mean when you bring it in Premiere the vid/audio goes out of sync? If that is the case, try demuxing video and audio with ffmpeg and bring them in seperatly.
To demux audio from an flv (or mp4) containing aac audio:
ffmpeg -i "path-to-input.flv" -acodec copy -vn "output.m4a"
To demux video from an flv (or mp4):
ffmpeg -i "path-to-input.flv" -c:v copy -an "output.mp4"
Or you could look into something like mp4box, an mp4 multiplexer (muxer/demuxer).
http://www.videohelp.com/tools/MP4Box
GUI for MP4Box:
http://www.videohelp.com/tools/My-MP4Box-GUI
An alternative GUI for MP4Box is Yamb: