(I'm writing this because I myself have heard this myth and at one point believed it could be the truth)
Myth - Doctors, EMS etc. will not do as much to save my life in a medical emergency if they know I am an organ donor.
Truth - All medical professionals are sworn to keep the Hippocratic Oath which basically states that they will do everything within their power to maintain the life of the person they are serving.
It makes zero practical, medical and ethical sense to "allow" one person to die (in this way) so that others might possibly live. Organ donation is never a sure thing for the recipient, and every single medical professional I have talked with about this subject (either on a professional or casual level) has given me complete assurance that they could not imagine anyone ever giving teeth to this myth. When you are in a medical emergency, their complete focus is on saving your life.
In fact, people who have agreed to organ donation are given more tests to determine that they are truly dead than are those who haven't agreed to organ donation.
Please, read more about Organ Donation Myths.
I'd hate to think that the only thing keeping somebody from becoming a donor is a lie.