We are spending a week in Paris and would like to know the best walking app to download on an Apple iPhone.
By "walking app," do you mean an app with guided walks, or a mapping app so you can find your way from point A to point B?
For the former, Rick Steves Audio Europe has several walks for Paris, and several for various museums and sights (including Pere Lachaise cemetery and Versailles). It's free. I'm sure there are other apps that have guided walks, but I haven't used them, and some are likely not to be free.
If you want a mapping app, Google Maps works fine in Paris for walking directions - at least it did on my Android in conjunction with T-Mobile roaming service. If you won't have cell service, let us know, so others who know how to get mapping apps to work without it can give the latest details (there are definitely ways, I just don't know them, and I don't know the ins and outs of iPhones).
For a mapping app that can guide you from Point A to Point B, I would recommend HERE Maps. Download the app and the map of France ahead of time on your phone, and you will be able to access it without using cellular or wifi signal while you're in Paris.
You can also go ahead with your account and save sites that you want to visit -- and your hotel -- ahead of time. Then you can just put them in your Here maps andfind out how to get around.
I use it in conjunction with CityMapper for public transport guidance (for this you do need cellular or wifi connection). I find it easier to follow the walking directions in HERE maps.