The biggest problem is that they were making insinuations that it was intentionally similar to her role in the movie Her (and even contacted her directly to do it).
There are laws for unauthorized use of likeness that could possibly apply here.
I'm not familiar with the specific laws that are relevant here, but I have trouble buying the argument that this constitutes a "likeness" when Johansson was in no way involved in its creation, other than by providing inspiration.
So they asked her and she turned them down. But guess what, they can get a pretty good result without even involving her. It turns out that her "talent" for the voice from the movie Her is in fact not so unique and valuable because it can be reasonably well replicated by other people, so why shouldn't a company be able to choose an alternative like that? A person should not have an exclusive claim to the rights of all voices that sound remotely like theirs.
Celebrities have some right to publicity, so in this case she could argue that they were watering down the value of her as an actress by widely selling an unauthorized sound-alike.
The laws around this aren't strong (which is why her response pointed out the need for legislation), and as I mentioned OpenAI would have been in a much stronger position if they didn't directly reference the film publicly several times. One could argue that they were using the film and her role to promote their product.
There are laws for unauthorized use of likeness that could possibly apply here.