We will be spending Christmas in Paris. Would anyone be able to recommend places that are open on Christmas Eve for dinner? Is anything open on Christmas Day?

Your best bet will be a hotel restaurant. What is your price range?

Earlier this week I was wasting my time reading menus on one site and one restaurant specifically said it was open on Christmas...I want to say it was La Christine, at 1 rue Christine.
Otherwise, there are probably great kosher restaurants in the Marais and asian restaurants all over.
Big city Paris...multicultural; even here in my little town in the US restaurants are open Christmas Day for dinner.

My impression from last year is that lots and lots of restaurants are open on Christmas day. You must reserve, however.

What's your budget (prices can be steep)?

L'Oulette was open last year. Reasonably priced, and they have a website with the menu. You must take a taxi.

We are looking for a moderate price. There are four adults in our family. The kids will be backpacking so nothing really dressy will work.

Regarding dressy...sometimes people will be more dressed than usual when it is a special occassion like Christmas..Like here in the US when you go out for a holiday meal, the men wear their ties, etc.
You might start checking with the hotels and restaurants that are possiblities for you and see about dress for that day.
Just to avoid feeling awkward.
Hotels always have a restaurant open, but make reservations!
(we were in Paris over Easter, it was very difficult to find a place for lunch)
Have a lovely Christmas~