As an immediate option, you can make an INFOPASS and go to your local USCIS office for a notation be placed in your Passport to the effect. Also the USCIS office can document the error in their system so when you request reentry the port of entry the CBP officer can see an explanation of the error on their computer system. Also, when you travel bring as much proof as you can showing that the LPR card you have corresponds with your other vital information and your attempts to correct the error.
In the longer term, you will have to request a replacement of the LPR card.
Legal disclaimer: The statement above is provided by CC Abbott is based on general assistance and not intended to be a legal opinion because not all the facts are provided. The person requesting information and all others reading the answer should retain an attorney who is permitted by the state bar within the jurisdiction who can examine the complete facts and provide a legal opinion on your case. All information provided in the above answer and other information provided by CC Abbott does not create an attorney/client relationship within any state of Federal law.
Go ahead and travel. When you return to the U.S., submit form I-90 to correct your birth date on your green card.
(213) 394-4554 x0 Mr. Shusterman is a former INS Trial Attorney (1976-82) with over 35 years of immigration experience. His response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.
I agree with Mr. Abbott and Mr. Schmidt. If your date of birth was submitted correctly to the USCIS, but was the victim of a typographical error, then you should be able to file the I-90 without a fee after having marked the box with the "card wrong due to USCIS error."
This answer is of a general nature and should not be relied upon as final, nor is it intended as legal advice. A consultation with an experienced attorney is always the best way to go.