Can a powered down cell phone be turned on remotely?

Who's to say that the phone is really off? If someone controls the firmware of the device then the off functionality could be replaced with state in which the phone appears to be "off" but is in fact maintaining a line of communication to a remote user.

However firmware cannot stop you from introducing a hardware switch to disconnect the microphone. A similar switch could be used to disconnect the battery. With physical control over the device you can just move to a lower layer than your attacker and cut them off.

A Korean researcher demonstrated this on Samsung Smart TVs at Black Hat this year. (Slide deck here.) He mentions that the malware was originally designed for cell phones, and that TV sets were even easier to attack because battery life did not give them away.

His basic premise is that if he owns your device, he owns the power indicators, too.

Remote power-on isn't a problem when it's never actually powered off.

As an example, iPhone alerts will wake up the phone even if it is turned "off" via the UI. The software is black-box and proprietary. With one of these common phones you have no assurances of anything.

Off has a different meaning now than it used to with respect to technology. There are different levels of power consumption: hibernate, sleep, deep sleep, off, etc. Ultimately, if there is power supplied (charged battery present) you don't really know what the phone is capable of unless you examine the source code of the software running on the phone and have an assessment of its hardware capabilities.