Conventional B&W films processed in the complete C-41 process will be ruined; the bleach step will remove the silver image, resulting in a completely blank strip of film. I know because the employees running a local CVS minilab swore up and down that they could process my B&W film a few years back. I'd not been doing much photography for a while, so I figured minilabs had progressed since I last checked. The employees then had the nerve to claim my camera was at fault when they handed me back my completely blank film.
Originally Posted by LowriderS10
That said, there is a grain of truth in there: If you process a conventional B&W film in C-41 developer, omit the bleach, use a wash or stop bath, fix it in a fixer (C-41 or any conventional B&W fixer), and then wash it normally, it'll be fine, provided the time and temperature of the C-41 developer was suitable for the film in question. In other words, B&W films are compatible with the C-41 developer, but not with the C-41 process as a whole.