I'm going to cut a close friend out of my life due to their mental illness
Why would you cut someone out due to mental illness?
They've been like this since I was a child. It is really irritating.
It sounds like you are being a bit selfish. Your friend needs your help and you should be supportive given their mental illness.
It's not my problem, I can't change them.
You cannot change, but you can always help. I am sure your friend would be disappointed if you knew each other as kids and were to now break it off.