What is the ruling of Islamic law on keeping a copy of the Holy Quran on my website, after the original website from which I downloaded this copy asked me to remove it, knowing that they do not own the copyrights to it?

The owners of books and programs have the right to protect their rights and prevent their dissemination, printing, or production except with their permission. This applies to the Mushaf (copy of the Quran) if effort has been expended on it that warrants preserving the right, such as adding an interpretation or special programming. If the Mushaf you downloaded involved clear programming effort by its owners, then they have the right to prevent its dissemination except through them. However, if the Mushaf is available on the internet and it becomes clear that its owners are not the ones who exerted effort on it and have made it available for free, then you are not required to delete it from your site, and you can download another available copy.