For the purposes of my project, I must use a 3.3v pro mini arduino to power it with a 3.7v Li-Ion battery. So I would like to know if using the Pro Mini at 3.3v has some disadvantages and what would they be?
The only real disadvantage is that it runs at about half the speed of the 5V version, but it is unlikely that you'd find yourself in a situation where that mattered.
I don't know how committed you are to the Pro Mini, but Pololu produce a version of the A-Start Mini that can work from 2.5-11V and runs at 5V/16MHz- it has an onboard step-up regulator.
One more difference I know of:
Half PWM frequencies
D3, D9, D10, D11: 245 Hz (490 Hz on UNO or 16MHz Mini)
D5, D6: 490 Hz (980 Hz on UNO or 16MHz Mini)
This may or may not bother you (most likely not).
I use 3.3v pro minis pretty often with 3.7 Volt Li-Ion batteries - never had any issues.
The big difference is that it can't be directly interfaced with devices that use 5v logic (nor even power them, since there's no 5v). The speed and pwm frequencies are also lower as noted above.
Make sure you use a 3.3v serial adapter, and if using serial adapter without the power line cut/disconnected, don't connect it to a battery when it's connected to serial (it doesn't have safe charging circuitry)