if there is no future and no prediction he can use "he seems ill" .
are you sure there is not any prediction or future meaning?
you see someone had a headache. and you can say to other friend.
he seems to be ill. ( i thik he will be ill, flu or sth. like this.)
and is not it used "he seems ill"?
As far as I know that in British English we can omit "to be" after appear, look and seem, as for example, "She seemed (to be) a good pilot.", but Americans do not say She seemed a good pilot. They use "to be"