Overcoming vbe in a low voltage application
Quick and dirty solution:


Replace the NPN with a PNP transistor or a P-MOSFET.
Swap non-inverting input with inverting input on the op-amp.
Add a 100 Ω resistor between the output of the op-amp and the base of PNP for safety. Skip this if you're using a P-MOSFET. 




Perhaps you can't acquire a PNP or a P-MOSFET, in that case just let the op-amp perform some math for you so everything will be fine.







simulate this circuit &ndash; Schematic created using CircuitLab

This way the op-amp will perform this expression, \$V_{motor}=V_{DAC}\$, if there wouldn't have been any math and you'd just swap the NPN transistor with the motor then it would've been \$V_{motor}=3.7-V_{DAC}\$.



EDIT

Andy Aka pointed out that that the schematic above will oscillate. So I tried that schematic out on a breadboard but with an N-MOSFET instead of an NPN and he was right. It oscillated at 180 kHz. 

However after attaching a 3.3 µF capacitor from ground to drain, or collector in the schematic above if I'd use an NPN transistor, the problem disappeared. But Andy Aka doesn't think it's reliable. 

With that said, don't use my design if your device that you are making has life threatening elements in it.

I won't delete this answer because the first part of the answer clearly says "quick and dirty solution".