All BBSs (single-line or multi-line) that I'm aware of require modems. And, also to my knowledge, there's no such thing as VoIP modem; you can attach a modem to a VoIP communications network (such as ShoreTel's, for example), but you'll still need to connect the modem to a land line; there's no way to do modem-to-modem communication via TCP/IP (again, to my knowledge).
The original intention of bulletin board systems was to allow people to communicate serially or in parallel (again, single-line vs. multi-line). That function has been pretty well taken care of by e-mail, bulletin board web sites, "Web 2.0" web sites, etc. There's really no need to have a BBS any more.
However, if that's what you want to do, there's nothing stopping you from digging up an old copy of WWIV BBS somewhere (I'm sure it's available for download if you look hard enough; looks like it's here, actually), slapping a modem on one of your physical COM ports, using a virtual machine to set up DOS, the BBS, and connection to the modem, then firing it up and giving your phone number out.
Actually, dial up modems were originally designed for, and used on equipment only intended for human voice. Looking at the ITU V.34 modem standard, you should be able to achieve between 14.4 and 33.6kbps transfer rates over a voip line. The other more recent modem tech may fail due to the fact that VoIP is different than a copper wire pair connected to a phone exchange. You are in luck that modem standards allowed auto fallback to the speed that is supported by both modem ends and the medium in the middle, though you may end up with as little as 1.2 kbps – BeowulfNode42 – 2014-04-28T12:21:02.163