There's a lot of factors. Temp, time, and pressure being the big ones. Dormant yeast will start to break down, releasing their contents into your beer (yeast autolysis), raising the pH and starting a savory, umami character. As it gets worse it can get brothy, then meaty, soy-saucy, and at extremes rancid or burnt rubber. It's most dangerous at pro level where the pressure of thousands of gallons of beer on top of the yeast makes it happen a lot faster. But it happens regularly in homebrew as well.
If your temp is good and you didn't, say, dump new wort straight on a previous yeast cake, a few days will be fine. A couple weeks after it's done and it can become noticeable (as much as many homebrewers on here insist it doesn't) especially if the temps are higher. A couple months, you're asking for problems. They won't likely be *intense* even at a couple months on the yeast (perhaps why homebrewers refuse to acknowledge it happens at all).
