Today it came to my knowledge that Intel intends to discontinue Kaby Lake-S processors on the 9th of October 2020.
I use an i5-7400 on my day-to-day desktop. Been using this CPU for 2 years. It does everything I need flawlessly.
Will I have any problems if I don't upgrade to a newer processor past the EOL date?
Absolutely not. Discontinuance just means that Intel will no longer be building these particular processors. You have a Desktop processor. This processor has a (3 year limited, I believe) warranty. In addition, Intel is typically providing driver updates for its processors for ~6 years (I do not know what they are publicly guaranteeing, however). Right now, graphics drivers are still being supported for 3rd gen processors (though support for Windows 7 has been dropped). Bottom line, you have nothing to worry about - for now, obsolescence will eventually catch up with you.