Designing a LiIon battery pack & charger for a Roomba 400.
My question is will a 4 cell, 14.4 V 4S Li-ion Lithium Battery 18650 Charger Protection Board 15A work to protect the batteries so I can use the standard charger that will provide 20 V to the charger protection board?


Choose one - listed with highest probability first:


No
Not really
Not very well.
Expect very short battery life.
Or very very short. 
Maybe for a while
Maybe 


A protection board is intended to be used to protect the cells against extremes and excesses. It is not intended to be a charger UNLESS that functionality is specifically included.
The example board includes the word "charger" in the description in some places and not in others. The specifications do not indicate that charging is a major intention.

Charger ICs and charging PCBs and whole chargers, targeted at 4S charging are 'common enough'. Design or use one.  

A protection circuit may not prevent overvoltage when charging, probably-won't terminate charging  when Ichg falls to a preset fraction of Ichg_max, probably does not limit Ichg to a safe level and probably allows discharge to voltages which are damaging in repeated use.