I mistakenly ordered a 4S instead of a 3S battery. Given the time it took to come, I am inclined to just keep it.

I have a couple of quads that I run on 3S and on which the motors are rated for 3S. Would I be buying trouble if I tried running one on the 4S battery? My APM2 on a separate BEC that can handle the higher voltage, and the ESCs should be Ok, but would I burn the motors?

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Your motors will run approx. 33% faster at full throttle which in itself shouldn't harm them unless maybe if they are really cheapos, and even then you probably won't be running a quad at full throttle anyway. Your throttle stick will be more sensitive so you may want to add expo and/or adjust the throttle curve in your Tx if it has those capabilities. If you fly at the same RPMs and with the same props as you did on 3S the overall efficiency may be affected somewhat, but it probably won't amount to much (but if you use a timer to determine when it's time to land, be conservative until you know how much of the 4S you're using per minute). Check motor temp by feel after a flight (a good habit to get into anyway). I've switched numerous helis and planes back and forth between 2S & 3S and between 3S & 4S and never had a problem (with appropriate BEC of course).   

Thanks. About what I figured, but I am still new enough at this stuff that I like to ask first. Guess I'll give it a try. Be interesting to see how it all behaves.

What motors, props, and payload?

If you have large props you will destroy the motors and escs due to your immense amp draw.

If you move to 4s you may want to get smaller props or limit your throttle to be safe.

If you have really good escs you may be fine, but its not really worth the risk of just guessing.

Find out what amperage your motors are specked to, and make sure you do not exceed that rating without healthy tolerance on your escs and even then be careful.

You will likely fly for slightly longer, ( higher voltage less current draw = higher efficiency.)

I have rotated before myself, but you have to be careful.

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