In most parts of the country this is an unqualified no.  However many municipalities have stricter rules than others.  I’m not aware of any place that has completely banned wood burning in fireplaces that were built before a certain time,  which are as they say “Grandfathered in”.  However several places like the Bay Area in California have banned any wood burning fireplace in any new home.  Washoe County Nevada, where I live, will allow you to put one in a new home, but with very strict limits of less than .1 grams per hour particulate emissions.  If your property is more than an acre, you can have one appliance which is EPA certified, which means it must emit less than 2.0 or 2.5 grams per hour, depending on the testing method. 

Even if you have a compliant wood stove, there are many places around the country that implement no burn days, spare the air days, or red flag days as they’re often called.  This is due either to increased fire risk in the area, or poor local air quality.  In that case you can get a waiver if your wood stove is your only source of heat, otherwise you will face potential fines for burning on those days. 

Check with your local fireplace dealer, they’ll have more information on specific local rules and regulations.  Also do consider your neighbors.  Your open hearth wood burning fireplace emits up to 69 grams per hour of particulate matter, which is known to cause many health problems.  http://burningissues.org/comp-emmis-part-sources.htm  Some of those particulates go outside to your neighbors, but some stay inside as well, reducing the quality of the air you’re all breathing.  Wood is a renewable, local fuel source, so if you plan to burn it, do invest in a quality EPA certified wood stove or insert.  Your lungs and neighbors will thank you, and you’ll also get more heat out of the wood you use (all those particulates are burnable material flying out of your fireplace).  That way you can be conscientious, and cozy.