There's a fictional mobile app in a Netflix TV show that I'd like to make a real thing. I'm an independent developer, I don't plan on monetizing it in any way, and it's going to be open-source. Is it legal for me to use the name, the logo and the color scheme that appear in the TV show? If no, how much of that can I use as-is and how much do I need to change so no one sues me?
The general idea of such an app is not subject to copyright protection. Ideas never are protected by copyright. So creating an app based on the functionality of a fictional app would not be a copyright violation.
The logo might, if it is original enough, be protected by copyright. Any or all of the "name, the logo and the color scheme" might well be subject to trademark protection. (Names and other short phrases are not protected by copyright.)
You would be wise not to use these identifying elements of the show, but instead create ones sufficiently different that no reasonable person would be confused into thinking that your app had been used on the show, or was sponsored, endorse, or approved by the show or its creators. An explicit disclaimer saying that you are in no way associated with the show or its creators, and your app is not approved by or endorsed by them would also be wise. Otherwise you might be accused of trying to pass off your work as affiliated with they show, or to trade on the show's reputation and fame.
Whether you make your app an open source work is not in any way relevant to copyright or trademark claims. Whether you charge for your app is of only limited relevance to a copyright claim. Whether you sell or market your app, or use it to advertise some other product or service is relevant to a trademark claim, as trademarks are only protected against their use "in trade" which generally means commercially. However, non-commercial use of a trademark may constitute "dilution" of the mark, which may give rise to a cause of action against the person using it.
A few years back, a developer made a free android app called Tricorder. The Tricorder is the name of a device in Star Trek. This app had a user interface that had a similar style to the computer displays in Star Trek: The Next Generation.
CBS's lawyers were displeased and issued a DMCA takedown request to Google. Google chose to respond by taking the app down from the Android app store.
Here's more information about my example.
This could happen to your app.
Is it legal for me to use the name, the logo and the color scheme that appear in the TV show?
How would that not be violating copyright? Read Is there a copyright on that content? and Is it legal to use scenes from famous TV show FRIENDS to create pop up museum in New York city?
I'm an independent developer, I don't plan on monetizing it in any way, and it's going to be open-source.
These mean very little in terms of copyright infringement, except in the possible legal defense based on Fair Use (Copyright Overview by Rich Stim - Stanford Copyright and Fair Use Center).
If no, how much of that can I use as-is and how much do I need to change so no one sues me?
A court would make the ultimate decision on how much and type of your use constitutes Fair Use.
And read Berne Convention (Wikipedia) for jurisdiction.