I can confirm that Handbrake will burn in sub-titles. I use this when there is foreign audio subtitle. TV broadcasts display these always but some files have them as a separate track. You can't guarantee that any playback device you use will access such tracks so I burn them in.Only certain types of subtitles are supported and then ONLY in certain container types.
I think that Handbreak can "burn in" subtitles (?)
I use TMPGEnc Video Mastering to burn in subtitles but that costs.