What is the ruling on someone who composes a poem mentioning that God created two Eid festivals, and that the third is the birthday of a certain person, and then posts it on a public website? And what should the website do about that?

If the poet intended that Allah has legislated a festival for Muslims on the birthday of a person, then this is a lie and a fabrication. If he intended that it should be glorified and celebrated like Eid al-Fitr and Eid al-Adha, then this is invalid, because festivals are among the rituals of the religion, and Muslims have only two annual Eids (al-Fitr and al-Adha) and one weekly Eid (Jumu'ah). It is not permissible to innovate any other Eids. Celebrating the birthdays of individuals is an innovation (bid'ah) and an imitation of the disbelievers. Therefore, the poet must repent and remove the poem from the website.