If you are a U.S. lawful permanent resident who has been convicted of a felony—or indeed any crime—then applying to renew your green card carries risk. You could end up being removed from the U.S. (deported). That doesn't mean you shouldn't try to renew the card. It expires every ten years, and you are legally obligated to carry a valid green card with you at all times. But in this situation, you should absolutely consult with an attorney before submitting your renewal application. Here's why.
In order to apply for a renewal of your green card, you will need to fill out and submit Form I-90 to U.S. Citizenship and Immigration Services (USCIS) and pay the appropriate fees for both the application and "biometrics." As you might know by now, USCIS's collection of biometrics data includes taking your fingerprints. After checking your prints against various databases, USCIS will find out whether you have any crimes on your record since the last time it reviewed your file.
If USCIS thinks your crimes make you removable, it will place you into the immigration court system, by initiating deportation proceedings. That means that you'll have to appear before an immigration judge and defend yourself against removal (deportation). You might argue, for instance, that your crime doesn't actually match up with any one of the grounds of deportability (described below).
If your argument fails, you will likely lose your lawful permanent residency and be deported from the United States.
The Immigration and Nationality Act (I.N.A.) contains a long list of types of crimes that make a green card holder deportable. (See 8 U.S.C. § 1227.) It's far more complicated than saying, for example, "You'll be deported with a felony on record." Some people who have committed a crime that is considered a felony in their state might nevertheless not be found deportable, while other people who have committed a crime that's considered a mere misdemeanor in their state will be found deportable.
For example, someone can be found deportable for having committed:
This is only a quick rundown of the federal law categories of deportability. To actually understand, for example, whether a state crime that was defined as a felony constitutes an "aggravated felony" for federal immigration law purposes requires an in-depth analysis of courts' interpretations in previous cases. Do not attempt to match the name of your crime against the above list without an experienced attorney's help.
If you have been convicted of a felony, yet need to renew your U.S. green card, hire an experienced immigration lawyer. In fact, look for one who specializes in the intersection between U.S. criminal laws and immigration laws.