CloudConvert now offers the possibility to burn text subtitles (hardsubs) into videos. Therefore you need the SRT file (SubRip Subtitle File) which contains the texts. Afterwards select the video file using the CloudConvert webinterface. As output format you can choose to keep the video format. Using the advanced options select the SRT file (see the screenshot) and start the conversion!
Update: We now also support the more advanced ASS files for adding subtitles!