Yes, it should. The phone line carries a small amount of current and should be enough to power an LED lamp. I wouldn't use this in anything but an emergency situation and then only sparingly. The system wasn't designed for constant draw of power, but instead has power in order to send it's signal and to ring you. I'm guessing that it degrades the quality of the line just by being turned on, and may prevent inbound calls from reaching you.
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Experience is a hard teacher because she gives the test first, the lesson afterwards.