There are benchmarks strewn all over the internet that seem to suggest that C++ is a much faster language simply because its much closer to the kernel than any other high level languages like C#. But these benchmarks are quite limited, in that, they are run over a small repetitive set of problems that seem to be computation intensive and unless you are building a scientific or mathematical computation system or heavy data system of some kind (like Facebook does), you won't need that high performance because it won't be utilised fully.
That being said, for your purpose which is building a server side program that will plainly echo a JSON string based on a request, C# will give almost comparable performance to C++. ASP.NET engine that runs with IIS uses native APIs written in low level C that are quite optimised for web servers AFAIK. But then again, every use case is different and every technology suits one set of problems more than others.
My recommendation is to start small and build 2, one with C# and the other with C++ and analyse the performance yourself over a period of time. At the end of it and based on your long term usage strategy, pick one that comes out at the top.
I will be glad to be corrected or added to.