If closed source software is distributed along with open source software, will the resulting code still be open-source?

A single repository can contain multiple, independent, copyrighted works. There is no requirement anywhere that the contents of one repository must be under a single license. And independent works have the property that their copyright licenses don't affect each other.

Based on that, there is very little against a repository that contains both open-source and closed-source software.

The main thing to look out for is if the two pieces of software are not independent works under copyright but one is a derived work of the other. For derived works, the licenses do affect each other and you need to ensure that the license on the base work allows the creation of derived works and that the licenses are compatible with each other.

With permissive licenses, there is little problem with license compatibility but it is definitely a problem with copyleft licenses.