I don't quite understand. Do you mean changing translation of the birth certificate? or getting a new birth certificate?
If it was honest mistake, human mistake happens. You must explain the circumstance.
If it was fraudulent document... well, it was fraudulent document thus immigration fraud/forgery.
It should not. It will create a red flag and a need to explain.
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The information contained in this answer is provided for informational purposes only, and should not be construed as legal advice on any subject matter.
I see you are asking this a few times. My work is primarily with refugees providing free services, such as assisting with applying for residency) to folks just like you and here's what I think:
If your case was already approved and you have the incorrect date on your green card, you can still attempt to get it corrected and it is unlikely you will run into any problems. If you have not yet been approved, then it is highly likely that Immigration will send you a request for evidence about the incorrect date of birth. You could try calling customer service to ask them to correct it, but this may or may not work. What would actually be best is for you to get a little help from someone who assists refugees in a similar manner as myself. If you were initially resettled in Madison, contact the agency that resettled you to see if they know of service providers. Otherwise, try contacting the agencies listed here to see if they can help you: http://www.immigrationadvocates.org/nonprofit/l...
What you are dealing with is quite common and you have a perfectly reasonable explanation. It is very unlikely that any kind of fraud will be attributed to you, but you need an attorney's or BIA accredited rep's help with "fixing" this. Contact some agencies and get the help you need or, if you do not qualify for any free services, consult with and retain an immigration attorney with past experience assisting refugees. The other reason you want to get some assistance is because what if you have other issues that need to be addressed that you do not know about? You can get legal information on sites like this, but this is not legal advice and the only advice you can actually rely on is legal advice (advice that can result in an attorney losing his/her license if it is wrong advice) and the only way to get that is to retain an attorney, a free one or one you pay. Good luck!