If I am using open source code in my project but don't want to release any source code, what license should I use?
Verify that the open source you are using does not contain a "copyleft"* provision, or any other condition that prevents you from using a closed-source license, and then use a closed-source license.
Note that the license you choose does not change the terms of the licenses of the open source code you are using in your project. If those terms say you can't close the source, then you can't close the source.
*Copyleft is the provision in the GPL licenses that requires you to release the entire source code of your project, if you use any software, code or libraries in your project that are GPL-licensed.