Yes, if you would put a resistor in series with the 5mm LED, the constant current output would just try to compensate until it gets 120 mA or until the supply limit is reached.
But you could put a resistor parallel to the 5mm LED, so that 20 mA flow trough the LED and 100 mA through the parallel resistor, which would make the current source happy. The question is, would that make you happy?
Do all of the LEDs have to be controlled individually?
Otherwise you could parallel up, let's say 6 of the 20 mA, LEDs and put them on one current output (but I would also give every LED a small series resistor to help with current sharing among the 6 LEDs).
Or you could set the TLC59116 to 20 mA output current and connect one of the 120 mA LEDs to 6 joined up 20 mA outputs. Or any combination of 6x20, 3x40, 2x60, 1x120.
EDIT: Also just want to mention that the outputs of the TLC59116 are capable of switching up to 17 V, so again if not all of the LEDs need to be controlled individually then you could put some LEDs in series and save some pins.