I watched an old episode of The Real Hustler in which they demonstrated how strangers can use some devices to force your Bluetooth-enabled mobile phone to call a premium rate telephone number. The victim's phone gets charged and the owner of the premium rate number gets the money. This TV show may have aired before wifi was common, so they did not mention problems with wifi. Assuming I have Bluetooth turned off, can attackers achieve the same thing using wifi? If so, would this only work when the phone is connected to a public wifi access point or could it also work while the wifi is not connected to anything?
In general, no. To a large extent, Bluetooth can be though of as a wireless replacement for the cables connecting your headset, keyboard, etc. to your phone; the hypothetical device involved works by imitating the phone keypad. Wifi is strictly a replacement for your network cable; your phone does not accept user-interface input from it.
That said, it's possible that specific smartphones have security vulnerabilities that permit a webpage or other network-delivered content to force the phone to dial a premium-rate number.