After some 'hummm, how might that be done' type thinking, it would very much depend on how and where the phone line was terminated at within the old analogue phone. In most of the ones I have dissembled the cord is pretty much run directly to the handset hang-up switch. If that's the case then I doubt it however a few that I've seen have terminated the cord on the main circuit board with some downstream components for various functions (call/line indication, remote ringing etc) before running up to the on-hook switch. If the phone is one of those .....

.. Then yes, I suppose in a James-Bondy way it might in theory be possible (with sufficient gain, filtering and DSP) to (ab)use an electrolytic capacitor, piezo buzzer (hard-of-hearing ringing booster) etc.

If it is one of the simpler ones, then no (but not a domain expert). Best suggestion is open it up and see how/where the cord terminates. If there is little between it and the cord coming in then I don't think there is any way to 'jump over' the air-gap created by the switch though I guess you might be able to measure the change in air capacitance of the switch air gap for very loud noises (spinning rust disks record loud vibrations/sounds) so as always, consult an expert based on threat model.

In terms of legal/type approval any testing most likely states that ‘phone must disconnect from line’ and be able to withstand x amount of surge before emitting magic smoke. Local laws may vary.

Note:VOIP phones are a different kettle of fish and haven't taken enough of them apart to comment on them.

I'm not familiar with the Ooma service or their devices, but if they're more or less using branded devices to do SIP, like Vonage and MagicJack and many others, while technically they could listen to the disconnected line, call audio isn't connected until after the phone is picked up on an incoming call or after the call is dialed on an outgoing call (dial tone is generated by the device at your home).

Like retrac said, a POTS phone is expected to disconnect the audio equipment when it's on the hook. That said, I had a particularly low quality phone growing up that didn't fully disconnect, and you could barely hear a conversation or, more usefully, call waiting beeps while the hook switch was closed, so it's possible the microphone may have been mildly connected or anyway speakers are also poorly tuned microphones, so maybe you could get something. I'd guess nothing salvageable after it went through a voip codec though, although maybe you could get an occupancy signal.

A standard PSTN phone disconnects the microphone/speaker from the telephone circuit when on the hook. Originally, a switch pushed open by the handset in its cradle.

In practice it would not be possible with just about any phone. Might as well be unplugged. But I wouldn't say guaranteed for all models. Too many ways to design a phone. It's not too hard to design an electronic switch to disconnect from the circuit that would have some faint crosstalk imposing an AC signal on the line even when it's supposed to be an "open circuit". Probably easier than designing one that truly eliminates it, especially back in the day.

Even if it can't, it is still a box sitting next to your phone and can have a hidden microphone in it