Given the decoupled architecture of DXM, there are two areas to consider where you might want .Net Core support:
- in the CMS (writing template and library code); and
- on your hosting environment.
In the case of in-CMS support, this seems unlikely as the bulk of .Net Core is really about runtimes and command line tools (see About .Net Core). .NET Core provides compatibility with .NET Framework by implementing the .NET Standard specification. What you probably want in the CMS is support for a later version of C# -- at least a version that is compatible with .Net Core.
On the hosting side, it depends on who you have your hosting with. If you are using your own hosting arrangements, then the question is moot because you can decide on exactly what you want. If you are using Crownpeak hosting, things are a little more tricky. Crownpeak provide both Windows- and Linux-based hosting, but crucially, from Crownpeak IT/Ops build packages / images. If you need to install anything, and this helpful article on deploying ASP.NET Core apps to IIS say you do, you are going to have to make a request to Crownpeak IT/Ops to support a custom build.
Could you provide some use cases where having .NET Core support would help your team? That way, Product Management can get better insight into what customers and developers would find beneficial.