Paris dinner attire

August in Paris is likely to be hot. If I and my spouse wanted to do this, I would have him wear good, new, dark jeans, or dark chinos, and a nice, solid color long sleeved shirt. I would wear good quality black slacks (not jeans) and a fancy, lightweight top (which should be easy to pack), with interesting jewelry, and black sandals. This would not be appropriate for a 3-star Michelin restaurant, but I think it would be acceptable in the nicer mid-range ones. If you find a restaurant that interests you, you can always call them and ask about a dress code, or ask at your hotel.
A few restaurants I like, that are casual and have charming decor: are Bistrot Vivienne (in the Gallerie Vivienne, near the Palais Royale), Cafe Ziti (a good Italian place (fresh pasta) on rue Pierre Charron, just off the Champs Elysees), Le Hangar (1 Impasse Berthaud, hidden charming place, in a small alley across from the Pompidou Museum), Le Vaudeville (29 rue Vivienne, near the Bourse) historic, classic French experience (my recommendation).

Good luck!

Oh, I forgot and Alex reminded me -- whatever you wear, don't wear white athletic shoes -- and you will be OK!

food is the only thing which has to be "chic" at the table
That's well said, Alex, and your English is just fine. Judy, nice casual clothing will get you into probably 98% of good Paris restaurants; dark jeans, and dark, clean sneakers for men are fine. If there is a dress code, it will state this in most reviews, Yelp etc. Not to worry! Have a great anniversary dinner!
By the way, we had a nice experience at La Regalade, on rue Saint Honore. Went on a Friday, walked over for the late seating after walking through the Louvre, which is open late on Friday's.

Wherever we travel, we both wear black jeans or slacks for my wife, black New Balance running shoes, one dress shirt for me, a couple tops for my wife plus scarves to accent and costume jewelry. We have been to fine dining establishments from Johannesburg to Rome, to Paris, to Buckingham Palace and tea at the Ritz and to the opera in Florence and Paris and have never received a second glance. Well, they did get a little perturbed at the Ritz but they got over it... or at least kept calm and tried to carry on.... I did wear a tie to Buckingham palace with a dress shirt and a tie is easy to pack along. We were in Paris last week and LOTS of people were wearing white running shoes plus all the bright colors so popular today -- bright blue, purple, green, orange with yellow laces -- just about any color you could think of. Those trying to style in Paris were easy to spot. They were the ones limping by early afternoon. Go and be comfortable first.

Yes, time the "no white running shoes in Paris" myth was put to bed. Who are these people who declare this to be true? Have they ever been to Paris? Parisians wear whatever they like on their feet and many of them like white running shoes.

Yes, many people wear white running shoes in Paris -natives and tourists alike. But the OP was asking about attire for a special dinner at a nice restaurant, and I think they may not be very appropriate there, (as Alex our French poster said) though some people might wear them to be sure. Black ones would be better. I have been to Paris 12 times and lived there for a year, and native Parisians would not wear them out for a fancy dinner. I never saw this.
PS I "style" in Paris, and walk for miles without pain (and am not young) and am preparing to do it again next week, and have never worn white running shoes. It can be done!

Wear simple black and a beautiful scarf. You'll fit in anywhere.