Both are correct but you would use them in different terms. Say for example you used to see your friend from school and now you stopped you would use "I don't see him anymore" and if you recently stopped seeing someone you would use "I'm not seeing him anymore".

I don't see him any more. ...or... I can't see him any more. When you are referring to what your eyes see then "see" is not used in the continuous. ...but... "I am seeing someone" = "I am dating someone".

I think the first one, give the context. If you saw someone, and then they disappeared. You'd probably say "I don't seen him anymore." But the second could work too somewhat