I did try the preview version of Core 2.0 on Azure Functions but it's absolutely woeful at the moment and the dev team confirmed they've done no performance work at all. As I recall the single request response time was around 15ms faster for Core than for full fat framework but the response to increased load was dreadful - never falling below a several second response time with the 400 concurrent user test and errors creeping in.
I largely set off down this path as even in isolation the Azure numbers were concerning and not at all what I expected from a serverless architecture and so I wanted a reference point.
As soon as I get an update from MS about Core performance work or it leaves preview I'll be rerunning the tests as I'm very aware that doing this across the two runtimes is less than ideal.
However from what I've observed the issues look to be more about the broader architecture used for Azure Functions than the framework runtimes. I'm hoping to learn more from Microsoft.