Many restaurants and bars will let you use their bathroom without having to buy anything, BUT NOT ALL. So I have listed a few possibilities.
Some metro stations have public bathrooms and they are usually very clean. These are some that I definitely know about: Metro stops on Rue des Ecoles, the Place de la Concorde, Trocadero, Pont Neuf. You will also find them at the Eiffel Tower, the Tuileries, le Jardin du Luxembourg, and in the park behind Notre Dame.
Other possibilities:
Quick, MacDonalds, Starbucks, department stores and libraries.
If worse comes to worst, you can use the free toilets you see on the street. I have yet to use one because I am claustrophobic.
At least one place I went to to use the facilities asked me for money. I had been there before and didn't remember paying but I gave the woman money anyway. The next time I went the attendant was not taking money so I asked her if it was free and she said yes. So evidently the person who had asked me for money was scamming me. Just to let you know it was the toilette behind the cathedral. So if they ask you for money there, they are doing so illegally.

