UPDATE for Christmas/New Year's 2015: Exclusive small-group Secrets of Paris Holiday Tours with Heather & Bryan.
Sure, most of the small shops and restaurants are closed, as well as a majority of museums. But you might be surprised to discover that there's actually a lot of things still open on Christmas and New Year's Day in Paris.
There's a French website that gives a list of museums open in Paris and Ile-de-France open on New Year's Day (open is ouvert, closed is fermé; noteworthy openings are the Musée d'Orsay, Centre Pompidou and the Musée Jacquemart André).
The Christmas markets are not open on Christmas Day and most close before the 31st, but you can visit the Grande Roue Ferris wheel, both ice rinks (from December 17th, at Hôtel de Ville, Eiffel Tower, and Charlety stadium), the free merry-go-rounds, the Eiffel Tower, and Parisian cemeteries (Père Lachaise, Montparnasse, Montmartre, Passy, etc.).
Most shops and showrooms on the Champs-Elysées are open, but may close a bit earlier. Shops and cafés in the Marais' Jewish district, mostly around the Rue des Rosiers, should be open as usual on Christmas Day (Hanukah is not a major holiday for the Jewish community in Paris).
The larger brasseries (such as those run by the Brasseries Flo and Frères Blanc) and restaurants remain open for Christmas. For the smaller ones, you simply need to to call and find out, since some are open and some aren't (in many cases they are either open for the 24th or the 25th, but not both). A lot of restaurants have special New Year's Eve menus so they may not be open on January 1st.
For entertainment, there are many shows. All of the standard tourist favorites like the Opéra de Paris, Moulin Rouge and Lapin Agile are open, but if you can read a bit of French you can look on a great discount ticketing website Ticketac.com for discounts on French shows including circuses, theatre, cabaret and dance.
Movie theatres are all open over the holidays. You'll find the English-language films ("VO") without French dubbing ("VF") at the cinemas along the Champs-Elysées and at the Forum des Halles, and all of the MK2 cinemas around Paris. With the smaller cinemas you'll have to check with each film to see what version they're showing (pick up a Pariscope at any newsstand, they come out every Wednesday when new movies are out).
Don't forget to check out the Secrets of Paris Calendar to see other events going on during the holidays.
Note for 2014: there are no fireworks planned at the Eiffel Tower for the 31st. There is, however, a light show at the Arc de Triomphe (Champs Elysées closed from 11pm) and a parade there on New Year's Day at 2pm (stret closed to traffic 1pm-6pm).
I've just added a new post for 2012, Paris Dining Options December 26th-30th.