What happens to ROS kinetic after the EOL date?

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Should you be concerned? If you'd like to see updates and fixes for your ROS packages, then yes.

However, there is quite some discussion around this. See [Poll] Kinetic users: Kinetic EOL in April, what is your migration plan? on ROS Discourse for instance. I write "however", as Canonical recently announced their ROS ESM offering: Hardened ROS with 10 year security from Open Robotics and Canonical, which may offer some relieve.

Kinetic also still uses Python 2, which has been EOL for quite some time now. Related discussion: Some things to know as Python 2 approaches EOL.

On top of that, Ubuntu Xenial itself is approaching EOL. Canonical's ESM offering may again help, but it's still not necessarily a good situation.

So again: should you be concerned? I would say: yes, unless you're in a position and willing to pay for extended support.

If not, you'll be using an end-of-life version of ROS, with an end-of-life version of Python on an (almost) end-of-life version of Ubuntu.

There really has been much progress since Kinetic, in many aspects of ROS (navigation, MoveIt, tools, support for newer versions of system dependencies such as OpenCV and PCL) that it's probably worth it to look into upgrading/migrating because of that alone.

If you have legacy systems to maintain, that's not always easy, so I'll be the last to say you must upgrade, but I'd strongly recommend to at least plan for it. You're already "almost" too late.