whether or not it stains depends on (A) the amount and (B) the concentration. If you're well-hydrated, a small leak won't leave a visible stain. Most large leaks will leave a stain, simply because the material acts as a filter and you get a "tidal wave" effect where the impurities for some reason get deposited on the edge when it dries, leading to peripheral / ring stains.
Best thing to do if you're worried it keep it wet until you can wash it. Once the urea dries, it crystalizes into a NON water soluable form and will trap the pollutants in a form that's very hard to remove with washing. (and this is also why it's so hard to get the smell of cat piss out of things) That's the idea behind the "wet pail", you don't let it dry until you can clean it.
I've woke up around 5-6am on a Saturday to a major leak, and as annoying as it was, I got up and changed, and took the sheet down and tossed it in the washer along with my jammies and started it.
If you can't toss in the wash immediately, find a pail and stuff the wet materials in and add several inches of water. You don't need to fill the bucket, just add enough that it will wick up into all the material and keep it all wet until you can launder it. You could even stuff it into a large (1 or 2 gal?) ziploc and add some water, that would make a fairly easy thing to set aside for later management.