Canada

Not... exactly.

Baseband processors could hypothetically remain energized and pull a list of nearby base stations, but it's not possible for it to be transmitting without you knowing (the battery drain would be extremely obvious). I have also never heard of this happening, nor can I find an example of this being documented behavior, nor can I find an example of someone enabling this functionality while the device has been powered off.

What can happen is something like the subversion of the power-off state to be simply an implementation of the OS's standby state - this is extremely difficult in any modern mobile OS (again I can't find examples of this being demonstrated without arbitrary code execution) but is the method rumored to be employed by the various security groups.

If you're being targeted by a group on the level of (ex:) the mossad, having a phone at all is a liability, even if the battery has been removed. If you're not the target of a security group, turning off your phone is a great way of preserving your privacy.

(Also, fun little side note, depending on your region leaving your phone at home won't do anything to prevent your location be tracked. Whee!)