All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Changing one’s real date of birth counts as a lie, and the basic principle is that lying is forbidden in Islam. However, if having one’s own rights depends on it and it does not incur any harm, it is permissible.
Ibn Al-Qayyim said in Zad Al-Ma’ad: “An example in this regard is lying to oneself or others when it involves incurring no harm to others if it would lead to getting one’s rights. This applies to the situation of Al-Hajjaj ibn ‘Illaat when he lied to the disbelievers in Makkah about Muslims to get his money back and no harm was incurred to Muslims.”
Accordingly, you do not have to change the date of birth in your birth certificate and document your real date of birth, especially if changing it would result in incurring harm or hardship. It is permissible for you to use your false date of birth only to claim your own rights. As for the benefits and interests to which you are not entitled and require having a different date of birth other than your real one, it is impermissible for you to obtain them with this fake birth certificate because this falls under the category of cheating and deceiving people.
Allah knows best.