I already filed my tax, but I realized I received 1095-B and my SSN is missing, the form only shows my birthday, and my school refused to update and send me a new form, will IRS ask me for a payment?
No, You should be okay to say you are covered by insurance as the form shows.
Just wanna make sure that the 1095-B is not supposed to be sent to the IRS right? So I will just need to keep it as my own copy?
No. It isn't attached to the return or mailed to the IRS. The school has to submit information to the IRS, but you are not responsible for that, as long as you provided the correct information to the school.