むり, むだ, and だめ, for when something's probably not a good idea

So, how about some vaguely related can't/shouldn't words?

無理 (muri), also written むり or ムリ

Though it's sometimes translated as "impossible", calling something muri doesn't necessarily mean that it's outright impossible, nor that it's necessarily wrong as such. Such an action does, however, go beyond the limits of what is normal, acceptable, or sensible. The word literally means "unreasonable", and often comes with a sense of pushing too hard, forcing an issue, going past safe limits, or simply ignoring what is realistic.

Expecting an untrained child to do the work of an experienced adult is muri. Staying up 50 hours straight to finish a project is muri. Pushing your starcraft's engines to warp 9 when they're not designed to go over warp 8 is muri.

It might be possible, it might get results in the short term, and it might even work out all right in the long run, but it's not going to be easy, and going through with it is likely to result in some unpleasant aftereffects.

無駄 (muda), also written むだ or ムダ

When an action is muda, it may be entirely possible and perfectly acceptable, but you shouldn't expect it to do anything useful. It's pointless, and a waste of time, effort, money, or other resources.

Trying to stop a tank with a butter knife is muda (it's also muri, for that matter). Hairs growing in unwanted places are sometimes called muda ke (they're not doing anyone any good). Buying a dozen donuts when you know you're going to end up throwing most of them out is muda as far as using money effectively goes. Resisting the Borg is muda (according to them, at least). Trying to beat the invincible version of Guardian in FF6 is muda (just like the game tells you). Building a lighthouse nowhere near a body of water is muda.

No one's necessarily going to stop you from trying, but you'd be better off focusing your efforts elsewhere.

駄目 (dame), usually written だめ or ダメ

Something that is dame may be expecting too much, like muri, or a waste of effort, like muda, or even outright impossible, but more to the point, it's just a bad idea. It tends to imply a value judgment, a sense that doing this would be, in some way, wrong. If you go ahead and do it anyway, expect there to be unwanted consequences.

Talking with your mouth full is dame (it's rude, potentially messy, and you won't be easy to understand). If your request falls on deaf ears, then asking was dame (the result was no good). Crossing a busy street without looking is dame. You're better off not even trying it.

The word may also be used in the sense of not being able to prevent something from happening any longer. If your outpost is being overrun by hostile aliens and you've run out of ammunition, your status is dame. When your bladder's about to burst and there's no toilet around, that's heading for a dame situation. Trying to keep a news story under wraps after it's already leaked is dame. Things have gotten to a point where trying any harder is likely to make them worse rather than better.

Objects and even people can also be described as dame; this basically equates to "hopeless" or "good-for-nothing" and may carry a sense of wasted potential or damage beyond repair.

For what it's worth, the term originated in the game of Go, where it refers to open spaces caught between opposing pieces that neither player can claim. As such, these spaces have lost any potential usefulness they might have had to either player, making them effectively worthless.