Has he told you why he rushes?  Like with homework... is he rushing to finish so he can move on to playing or some other preferred activity?  It might take you and your DH to actually sit with him as he goes through every homework assignment and rein him back every time he starts to rush through.  You could also work with him on reading comprehension exercises and word math problems.. they will be the ones that will be most at risk if he rushes through.

Teaching him techniques like reading through the whole test before starting to answer it... and reading questions more than once before answering are also good.

It's not really going to be good to punish "rushing" in particular... I mean, if he rushes because he knows the material well and can move quickly through an exercise.. that in itself isn't a problem... it's when he doesn't that it bites him in the tail.  So you punish the results.. go over the material after the grade and see if he could have scored better with more thoughtful approach and show him the difference between the A and B.  Perhaps even incentives when he performs well on things will motivate him to be more thorough with his work. 

If he is rushing so that he can go do another activity, then start setting a limit on how much he can do said activity in the evening or set it to beginning at a specific time provided his homework is done. If he thinks rushing means he gets to play extra on video games or whatever, then say he only gets an hour each night, or that he can't play before 7pm but ONLY if all his work is done. If he finishes early, then let him read, do an art project, get some exercise - something fun but not mindless.

Also, if you make him do it as soon as he gets home, try changing it up a bit. Let him get a snack and play outside or read or whatever for 30 minutes. Then have him do 15 minutes of homework. Then another 10 minute break. Then 15 minutes homework. Wash, rinse, repeat. This worked well with YSS as he would either procrastinate or rush because he didn't want to have to sit for 45-60 minutes. Breaking it up helped him visualize a goal, and he got done much faster with better results than making him sit and do it all at once.

I hate to be the bearer of bad news here, too, and say that nothing you do may matter. He may just truly not care and won't change his behavior. There isn't anything you can do about that, and you'll drive yourself nuts trying. Natural consequences, such as being held back or being prohibited from a club/activity, or not being as "smart" as thr other kids, may correct the problem - or it may not.

Like ESMOD said, you need to figure out the reason why he is doing it and tailor your approach based on his response.

I have no clue, but would like a solution too!  My BS8 is very competitive at literally everything, including school work.  So he will rush through his work so he can be the first one done. His teacher has mentioned it, but is not terribly concerned about it, so I don't put too much weight on it.  I just worry that as he gets older if it continues it's going to cause problems with his grades. Right now his rushing is not affecting his comprehension at all and his grades are great, but I think it's because he has a great teacher who loves him and if he scores poorly because he was obviosuly rushing, she will have him make corrections and hand back in.  But not every teacher is going to do that!