Should you take iron supplements while on birth control?

Not necessary. There's nothing specific about birth control pills that requires that you should take iron supplements. There's no problem doing so if your doctor has told you that you need them. If you were having heavy bleeding as a reason to be on birth control, iron may be needed.
No. You only need an iron supplement if you have heavy periods or have been instructed by your doctor that you have iron deficiency anemia. One benefit of birth control pills is that they usually make a woman's period lighter and/or shorter. Also less cramping.