Yes, they are always open. There are many cemeteries within the city of New Orleans that have very interesting histories and some with some famous people who have been interred there.
At the St. Louis cemetery Nicholas Cage has a pyramid like spot where he expects to be placed when he goes on to his big reward. The famous voodoo queen Marie Laveau has a tomb where people still come and make offerings and make wishes hoping that she will help to fulfill them.
All of these cemeteries are open but not all the time. They close at night. Some close periodically for maintenance. The one across from the famous restaurant Commander’s Palace, the Lafayette Cemetery is a popular one to go to.
You will see in these cemeteries above ground tombs which are a unique Spanish custom. There are neighborhoods in different parts of uptown where you will find cemeteries as well.
You can also take cemetery tours which are very popular with visitors!