Maybe. It assumes that you know whom the sperm donor is. And even then, it’s not so clear.
We have a general rule that children born in wedlock are assumed to be sired by the woman’s husband. While it’s true that genetically the child may not be the husband’s, it may be so for halachic reasons. Clearly this is an issue to be asked to a rav.
In the event that the sperm donor is not known, you can probably go via the principle of rov — since the majority of men are not Kohanim or Levi’im, you could probably safely assume that the child would need a pidyon.
If the sperm donor is known, then even if he is a kohen or a levi, then you would probably still need to ask, based on the above.
If the sperm donor is the husband, well, then, I guess there’s not much of a question.
The Wolf