In theory I suppose it is possible, but it would be totally impractical. Typical GPON maxes out at around 6dBm (4mW) of transmit power, passively split 32 or 64x, so the ONT receives only maybe -10dBm, which is 10-100s of microwatts. Even if you could harvest 100% of that optical power (you won't come close), it's not enough to be useful.
If you wanted to throw more power for this purpose alone, not only would you need relatively expensive filtering / splitting optics at the receiver to protect it, you would run into eye safety issues. Anything running into a layman's home is going to have to be intrinsically eye-safe (Class 1), so the maximum power is quite low, maybe 1mW. Still not very useful, even ignoring the insertion loss of the cabling system and the cost/safety problems of the many-watt laser you will need at the head-end for each 32 customers.
And you can expect that anywhere you are going to run this is going to have local power almost all of the time anyway, so this would only be used very rarely. It makes far more sense to just provision batteries at the ONT, and when fibre services are replacing copper, at least around here, that is typically part of the default installation to provide a few hours of standby power for the phone lines.