Probably I am not the first one to ask this question, but I can`t find the answer. Let`s say that I use the same game configuration and the same screen resolution. If I pass the refresh rate from 60 Hz to 85 Hz, do I lose some game frame rate (or game performance)?

It depends if V-sync is enabled. If it's enabled, it will limit your FPS to your refresh rate, so yes if your refresh rate is 100hz for example, your max FPS will be 100FPS.

But if you disable VSync, there's no FPS limit. (but anyway can your eyes really see the difference between 85fps and say 95fps? :o )