Best resistance for RED/BLUE 3 lead LED
A bi-color LED with three leads is just two LEDs sharing either a cathode or an anode. If you want only one resistor, then you can have only one color on at a time. You might do something like this:







simulate this circuit &ndash; Schematic created using CircuitLab

If you try to turn both on at the same time, then only the one with the lower forward voltage (red, in your case) will turn on.

If you exceed the maximum current for an LED, it overheats and is damaged. Less current just makes the LED less bright, but does not damage it. So, the thing to is calculate the resistor you might use for each one individually, then use the biggest resistor of all the options.

If the difference in brightness is unacceptable, or if you need the ability to have both colors on at once, you must have two resistors.