This question comes up a lot and the simple answer is , it depends on a lot of factors ...
If you started with a sterile wash and kept it under airlock at room temp for the entire duration of the ferment ...well, it's possible it could go for a month (or more) without going bad .
More times than not, the wash didn't start out sterile, and maybe we coulda used StarSan on the fermenter instead of eye-balling it and calling it good. Or maybe the temp's been in the high 80s for this wash...and we all know how bacteria like warm weather. ...Prolly opened up that fermenter and peeked in there once or twice too, didn't ya?
Beer can last a long time in the fermenter because it has hops in it that act as a preservative. Wine usually has sulfites added after a couple weeks of fermenting to preserve it.
Many of the popular recipes require no boiling because they're meant to be run right after fermentation ...so as a result, your wash is loaded with bacteria that is usually held at bay by the yeast activity during the ferment. The longer it sits after the ferment is complete, the greater the chance of spoilage.
I won't start a wash unless I feel I can run it in a week, (give or take a couple days) ...I don't think I've ever had a wash sittin around for two weeks, but I feel pretty confident that I could let it slide if I needed to. But If I had to keep a wash beyond 2 weeks, I'd refrigerate it.
In the end, if you couldn't run it for 3 weeks and you're worried it's gone bad, open it up and smell it, taste it ...if it smells reasonably ok, run it ...then decide. Because it's amazing how a sour corn bourbon wash can make you go , yet the distilate is
Hope that helps