Not neccesarily.
The phone landlines draw power from the electrical distribution lines via a small transformer placed here and there, but at each station, there are batteries which are kept charged, so if the electricity goes off line, the phone lines still have power for a while. No idea for how long, depends on usage and battery amp-hours, but likely a day at least; phones don't draw all that much power.
If you kept the phone line from the pole when you switched to cell, you can still use it by plugging in your old phone, but only to call 911.