As far as I'm aware all modern phones will at least partially wake up to sound an alarm even when turned off/

Does the iPhone/Android phones do this?

The iPhone alarms do not work when the phone is turned off. If you've muted your phone this also means that all alarms are muted too.

May be useful in business surroundings, but this makes the iPhone an awfully bad travel alarm.

Mute doesn't silence alarms on the iPhone, however the volume control does work. 0% and you won't wake up.

Oops, you're right, they do work when your iPhone is muted.

My really cheap Nokia will do this, even if the battery is dead enough that the phone won't turn on normally.

Blackberries do this.

"Wake up"??

If my Android phone is off, it's off. It doesn't turn itself on, period. If it's just idle, screen off, with lots of stuff asleep, sure it will "wake up" in that the screen turns on, and sound an alarm. But that's it.

That is not true at all for Android. Ask any Android developer. it is entirely possible to code an alarm callback that will not activate the screen at all, but can use a network to send and recieve data.

This is done commonly by builtin services.

That's generally what's meant by "wake up." This would seem to indicate the phone's not really "off," though I couldn't say how possible it is to tap into the mic in this state.

It isn't a modern phone, but I remember my Nokia 2600 (not Classic) doing that.

My couple months old Nokia C7 does this (but I'm more grateful for the convenience than worried about covert tapping devices..)