From what I can tell, it might, but Metro PCS i think is going to drop CDMA in late 2015, which will then render the use of the Verizon Note 3 useless for 3G and voice.
It should work in gsm mode. The Verizon Samsung Galaxy Note 3 is factory sim unlocked and is quad band gsm. You might not get all the bands though. You would need to set it to gsm mode.
I was able to get 4glte on tmobile on my Verizon Samsung Galaxy Note 3 in lte mode, so you should be able get 4glte. It doesn't support utms 1700 so you would only get hspa+ in refarmed areas.