I have a set of RNAseq data for 24 samples, half of them human culture, half of them human mixed with mouse.  (Also, three replicates each with 4 treatments) For the mixed ones, about 50% of the counts map to mouse genes.  I don't believe the submitters care about what's going on in the mouse genes.

Should I put mouse and human counts into DESeq2?  Or just human counts? 

This support site is, in general, intended to help people with technical problems that may arise when using Bioconductor software. Your question has more to do with how one would analyze a particular set of samples, which is beyond the scope of the site. By asking people on a random support site you are in essence off-loading the responsibility for the analysis of your data to strangers who know nothing about your experiment, for which I would offer the following advice.

1. Don't do that
2. If you don't know how to analyze your data, find somebody local who does, and give her money or %FTE or whatever. This is a non-trivial task, and just because you have access to the tools doesn't mean you have the skills. You can buy scalpels on Amazon (I presume), but shouldn't do surgery at home, and the same applies here.