If you are a U.S. lawful permanent resident and have been convicted of a felony -- or indeed any crime -- renewing your green card will put you at risk of removal from the U.S. (deportation). 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 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 must 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 may 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 that your crimes make you removable, it will place you into removal (deportation proceedings). That means that you'll have to appear before an immigration judge and defend yourself against deportation, very likely by arguing that your crime doesn't actually fit within 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. It's far more complicated than saying, "You'll be deported with a felony." Some people who have committed a crime that is considered a felony in their state may 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.