There are two problems:
1) If you bought it from T-Mobile, it's almost certainly locked so that it will only accept T-Mobile SIMs. You'll have to get them to unlock it, since I believe that it's currently illegal to do this yourself in the US.
2) Verizon are a CDMA network, and a T-Mobile HTC One (GSM) won't work on VZW. Different radio interfaces, so the Verizon One has different hardware from the T-Mobile version.
So if unlocked you could use it on another GSM network (but check 3G/4G bands for compatibility), or use a virtual mobile operator's SIM in the handset. But it won't work on Sprint or Verizon networks.