Can a Verizon iPhone 5s use T-Mobile LTE?
Short answer, yes (assuming you have LTE coverage in your area by T-Mobile). But you should get the iPhone 5s from Verizon instead of getting it from T-Mobile. Also, carrier unlocking has nothing to do with 3G or LTE.

Long answer follows below.

The iPhone 5s models A1533 (GSM) and A1533 (CDMA) support the LTE bands that both Verizon and T-Mobile use. 

If you look at the technical specifications for these two iPhone 5s models (A1533 GSM and A1533 CDMA), you'd notice that:  


The LTE band coverage is the same across these two models
The GSM model supports only GSM (so you cannot switch to Verizon even if you want to, since Verizon is a CDMA carrier)
[Model A1533 (GSM): UMTS/HSPA+/DC-HSDPA (850, 900, 1700/2100, 1900, 2100 MHz); GSM/EDGE (850, 900, 1800, 1900 MHz); LTE (Bands 1, 2, 3, 4, 5, 8, 13, 17, 19, 20, 25)]
The CDMA model supports both CDMA and GSM
[Model A1533 (CDMA): CDMA EV-DO Rev. A and Rev. B (800, 1700/2100, 1900, 2100 MHz); UMTS/HSPA+/DC-HSDPA (850, 900, 1700/2100, 1900, 2100 MHz); GSM/EDGE (850, 900, 1800, 1900 MHz); LTE (Bands 1, 2, 3, 4, 5, 8, 13, 17, 19, 20, 25)]


So you would be better off going for the iPhone 5s CDMA [Model A1533 (CDMA)] on Verizon and then move to T-Mobile.

Also see:
The Verizon iPhone 5s comes unlocked just like the Verizon iPhone 5