Everything that's running on your PC will make use of some resources, which means your game won't be able to access those resources. However, if your game doesn't actually need them, it won't be affected much if at all.
So it all depends on how powerful your PC is, how much (and what kind of) resources the game needs, and how much (and what kind of) resources are currently in use by other processes.
There are 3 ways a program can affect a game's framerate. In no particular order:
- RAM (Random Access Memory). As long as there is enough RAM, your framerate won't change at all. If there isn't enough — even if only by one Byte — it can significantly affect your framerate, either all the time, or only in specific situations.
- CPU. If you have more cores than the game can use, and as long as there aren't too many CPU-heavy applications running, you should be safe.
- GPU. Usually has the most immediate effect on framerate, but isn't used by many programs (especially not ones running in the background).
In most cases, apps running in the background shouldn't affect gaming performance much, unless your PC is very weak and barely meets the game's requirements. Browsers are a bit of a special case, however.
The best way to know if your browser is impacting your framerate, is benchmarking: Run the game both with and without Chrome running, compare the performance, and decide whether the perceived loss of performance is acceptable or not.
Alternatively, you can open the Task Manager (right-click on your Taskbar -> Task Manager) and see how much resources your browser currently uses. In my case, it uses between 1 and 2% CPU, 0% GPU, and a little above 3GB Memory (RAM). That's a lot of RAM being used, but since my PC has 32GB to spare (and most games don't ask for more than 8GB), my browser's impact on gaming performance should be negligible.