You may 'turn off chat' for certain people on Facebook. You still receive their messages but they no longer see you as online.
How do you turn off chat for someone who is not your friend on Facebook?
>>17499598
You block them.
>>17499598
If you aren't friends then the messages wont go into your inbox, they go into message requests.
If you don't want someone messaging you, block them.
>>17499603
This. Why would you still want to receive their messages?