The short answer to your question is yes, you can lose your green card. We assume you're not talking about literally losing the card itself, such as dropping it on the street -- in which case you'd want to replace it. Physically losing the card doesn't mean losing your permanent residence, it just means you've lost the evidence of that residence.
But you can also lose your permanent residence, for any of a variety of reasons. Although a green card entitles the bearer to many benefits in the United States, not the least of which is to live and work in the country legally, these rights come with limitations. Failure to know and abide by the rules can lead to your losing your residence. Even if you're not "caught" in the short term, applying for U.S. citizenship may bring the problems to light -- meaning you could remain stuck in the somewhat insecure status of green card holder for a number of years, afraid to apply for citizenship.
By way of a short summary, you can lose your green card by doing any of the following actions. These come from the grounds of deportability in the U.S. immigration laws. The laws themselves are written in much greater detail, so do not make any final decisions about your immigration situation based on this list -- see an attorney.
If you appear to be deportable, the immigration authorities may place you in removal proceedings, and ultimately remove (deport) you from the United States. For more information on removal proceedings, see "Overview of U.S. Deportation/Removal Proceedings."
U.S. citizens are granted more rights than lawful permanent residents (LPRs). They are allowed to work in sensitive government positions, law enforcement, and other restricted jobs that LPRs cannot. Citizens can also vote and serve on juries. If an LPR misrepresents his or her status as a U.S. citizen in an effort to vote, he or she can lose his green card.
Be particularly careful when renewing your driver's license. Under what's known as the "Motor Voter" Act, people can register to vote when obtaining or renewing their drivers' licenses. But the motor vehicle officials don't always know the difference between a USC (United States Citizen) and LPR, and may encourage you to register even if you're not eligible. If you then use your driver's license to vote, you may become deportable.
When you change your address, you must report this change to U.S. Citizenship and Immigration Services (USCIS) within ten days. The method for doing so is by filing an AR-11 Form, Change of Address) with the Department of Homeland Security (DHS). Keep a legible copy for your own files. The form can be downloaded from the USCIS website at no charge (www.uscis.gov). Better yet, you can submit the form online.
You're not just allowed to establish a permanent residence within the United States once you acquire a green card, you are expected to establish a residence and live there. At some point, you may be asked to prove this, for example when applying for U.S. citizenship or requesting reentry to the U.S. after a trip to another country. The best way to do this is to:
If you commit any of various types of crimes (it need not be a felony), or if you are caught with a controlled substance, you may be subject to deportation and the loss of your green card. Exactly which crimes make you deportable is difficult to summarize; this will depend on the facts of your case and the way the law is written in the U.S. state where you were convicted.
If you have committed any fraud in the application process for a green card and it is discovered, you can lose it or be denied the green card. For example, entering into a sham marriage to a U.S. citizen would be considered grounds for removal.
If you believe that you may have become removable from the U.S., or are arrested for a crime, consult with an immigration attorney immediately. This is important even if you already have an attorney representing you in a criminal case -- very few criminal lawyers are expert in U.S. immigration laws, or understand the consequences of filing certain types of pleas on your immigration status.