Yes. Please do not do this. A couple months ago I helped rod out kitchen plumbing clogged with coffee grounds. You really don’t want to do this. It’s icky. It’s a mess. Seriously bad idea.
Put them on your compost heap or in the garbage. Never down the drain.
I gather some folks haven’t actually seen a french press in use. Here’s a youtube video demonstrating how to clean out a french press (whodathunk you’d need an instructional on that). Notice demonstrator dumps grounds right down the sink drain. This is also how I do it as well.
I also ground up eggshells in my garbage disposal because I’m told eggshells have an abrasive quality that helps clean the inside of the pipes. Am I going to hell? 
IMHO, that guy is using about 2/3 as much coffee as he should be. 
You think 2/3? Admittedly, that’s the first thing I noticed and I immediately wondered why a guy who drinks coffee-flavored water is giving a french press demonstration. Since you mention it, I go back to do a little more research and…OMG, what a neophyte!
He’s starting out with pre-ground beans! :smack:
He’s using about 6 tbsp for what appears to be a 12 oz press (actually that’s pretty accurate measurement; even coffeegeek agrees. But still I use 8 tbsp; it’s much more flavorful and you can’t see through it (I believe see-through coffee is an abomination of epic proportions).
Then again, he advises viewers using whole bean coffee to grind beans longer for more flavorful coffee. :smack::smack::smack: No, no no! A french press requires a coarse grind lest you get a cupful of sludge. I bet he’d recommend a blade grinder for that. What a maroon!
I can’t watch the next installment. I fear he’ll use water he’s brought to a rolling boil. I’m already getting a little lightheaded. Damn you, Knead! You’ve agitated the coffee snob in me. Can I buy you an espresso sometime? 
That’s pretty much how I do it, too. I rinse out the grounds, then I add a drop of dish detergent, some water, put the slightly-loosened plunger back in and pump it up and down a few times, and then rinse it well and let the whole thing air dry. If the dishwasher is ready to run, I’ll put it in there instead.
So do I. As with coffee grounds, I’ve never had a problem.
We’ve gone over this topic a few times over the years. I think the problem is that most people don’t let the disposal run long enough or don’t use enough water. I never put grease or bones or stringy things like celery or onion skins in the disposal, though.
You should see my mother’s coffee. It’s so light brown, it almost looks like weak tea. She claims to not like strong coffee, but always compliments my FP coffee. Go figure.
All us French Press snobs KNOW you need a coarse, chunky grind! 
Damn, it’s 11:30, and now I want a cup of coffee…
My in-laws make the same stuff. That’s the problem with old people. They don’t know how to make good coffee, because they murdered their taste buds a long time ago on has-sat-on-a-hot-plate-for-hours-in-a-glass-carafe diner poison. 
Hay, if the coffee leaves a white cup white or the spoon tips to the side of the cup then it is to weak.
If we keep meeting in coffee houses, people are gonna start talking.
I keep an old drip cone (the type you put on top of a coffee cup) in my sink with a metal mesh filter in it. I add a little water to the grounds in my press, swirl, and then dump it into the filter to drain before throwing it in the trash.
I live in one of those older houses with some strange retrofits to get plumbing to the third floor apartment.