There may be some programs, which will hardcode subs into .avi file, but as far, as I remember, try to solve same issue, it was too complicated.
You don't even have to burn actual DVD.
Burning program, converts .avi file to DVD format, first, creating a folder video_TS with .VOB files in it.
Just point ripping program to video_TS folder and the program will do the rest.