Taking a callboard to a virtual set-up seems to introduce too many routes of potential failure, not to mention requiring a lot of accessories that complicate an otherwise simple process.
Let's look at it point by point:
Setup: Callboard - find a wall, stick callboard on wall. Add papers as necessary.
Virtual - find web space (usually requiring payment), write web code or configure existing code, send out log-in information (easily lost or forgotten), train all parties to use web space, find electric and web access in all locations in the theatre where site will need to be accessed, 90+ day learning curve to get people in the habit of checking something new.
Mobility: Callboard - take down call board, move to new location. Explain new location to company. Information cannot be accessed if people are not in the theatre.
Virtual - can be accessed anywhere, provided that people have either laptop, ipod, cell phone, etc. - not guaranteed.
Security: Callboard - if mounted in a room that can be locked, security is pretty much ensured.
Virtual - if it is online, there is NEVER a guarantee that other people will not see the data. Your data is housed on a remote server somewhere in the world, maintained by strangers and subject to many outside forces.
Dependencies: Callboard - needs wall, mounting method, pushpins, pen.
Virtual: needs computer access, electricity, internet access.
Legality: Callboard - certain labor and union mandatory documents MUST be posted in an area visible by all employees, i.e. workman's comp and AEA contact information. Callboard counts. Virtual callboard - I don't think there is any precedent for this.
Other uses for call board that may not translate well to the web: Signing in for call times, rehearsals, etc. Checking what's next on the rehearsal schedule *during* rehearsal.
The web is a tool. It is a wonderful and versatile tool, but it does have its limits.