Praise be to Allaah.
There is nothing wrong with using the umbilical cord or the placenta for medical purposes.
We put this question to Shaykh Muhammad ibn Saalih ibn Uthaymeen (may Allah have mercy on him):
What is the ruling on keeping the placenta to treat cancer and remove wrinkles from the face?
He replied as follows:
It seems that there is nothing wrong with it so long as it is proven (to be beneficial).
Question: Does the principle that whatever is cut from a living being is dead meat' (maytah) apply?
Answer: The maytah of a human is taahir (pure).
Question: If it has no use, does it have to be buried or can it be thrown anywhere?
Answer: It would seem that it is similar to nails and hair - i.e., there is nothing wrong with either throwing it away or burying it.
And Allah knows best.