Unfortunately, we do not have sufficient information from you for a final assessment: Do you still have a valid Iranian passport or are you already a German citizen entitled to a German passport? Which residence status do you have?
In general, the data in your passport must be correct. Otherwise, you can get into trouble, i.e., you could be fined. Correcting wrong data in identity papers is difficult and lengthy.
However, your right to reside in Germany is only endangered if you have given false information about your identity. Then, the residence permit can be revoked. But this is not the case with you. You did not deceive about your identity, there were just two – relatively insignificant – details about your date of birth missing in your passport.
It is good that you have a translated and certified birth certificate from Iran.
In principle, the Federal Office will only correct incorrectly recorded personal data during registration if original identity documents are presented later, the authenticity of which is also established after a check. It therefore makes no sense to apply for the correction of personal data such as the date of birth without submitting documents. Now, the foreigners’ authority requires you to present an Iranian passport, apparently the birth certificate alone is not enough for them.
We would recommend you to talk to your foreigners’ authority and to clarify your residence status in general. Probably, you are even entitled to an unlimited settlement permit.
This article has already been published in German.