Definitely get him checked medically as others recommend, to rule out anything that is really beyond his control. Be sure that he does not view the medical checkup as a form of punishment or some way of embarrassing him into doing what the adults want -- he will resist more if he feels "Mom's dragging me to the doctor over chewing?! I could die!"
Then, if there is no breathing issue or other medical issue, sit down with him when
--It is NOT a mealtime or close to mealtime
--Things are calm and he's not on his way out the door to somewhere more interesting
--His siblings are nowhere nearby
And talk with him, treating him like the big kid and a responsible human. Tell him, "Hey, I know we've talked at meals a lot lately about chewing with your mouth open.I know it bugs you to have me correct you a lot, I do get that. I wanted to ask you, now that we're not sitting at a meal this minute, what do you think is going on with that? Is it maybe because you feel you need to eat fast to get finished and go on to other things, or you just don't notice it? I'm not mad here or anything, I just want to know how we can work on it together better because it's important." Then wait. Really listen. He may get sheepish and say, "I dunno" and try to avoid it. But at least make a stab at approaching it as "I know you don't do it on purpose! But we do need to work on it. What ideas do YOU have that could help you stop doing it?"
That gives him some sense of control. Kids his age need to start getting a little control and being asked what they think they need to do something differently. Again, he may balk or clam up but maybe he'll engage with you.
And yes, the ideas like removing his plate, or asking him calmly to leave the table (AFTER one or two warnings first) can help.
I want to add, do keep working on this with him. It's worth it. Our niece is pushing 15 and chews horribly, mouth open, smacking. Her overall table manners are gross, period, and it's entirely because her parents have never corrected her at all. I dread the thought of this girl on a date, or eating a snack on her break at a job someday where a boss sees her, or heaven forbid, having a job interview over a meal one day.....