I know that white wines don't age, at least not like red wines, but do they get old? If there's no reason to leave them in your wine cellar for years, is there reason to drink them sooner?
Like some humans, many can, indeed, age gracefully. But most just get old quickly.
White wines lack or are very low in the tannins that act as antioxidants in red wines. So, they're not as protected against the ravages of oxygen. But a small minority are prized for their ability to improve with cellaring. Examples I'd lump into that category include riesling (especially German riesling), fine white Burgundy, vintage-dated Champagne, Australian semillon and Sauternes. In fact, the last in that list ranks among the longest-lived wines, period, capable of improving with many decades in the cellar. Acidity, generally higher in white wine, and sugar act as preservatives, too.
In fairness, aged white wine is something of an acquired taste. I wouldn't recommend it to most people. It can develop odd flavours that many detest. I'm talking nutty tang along the lines of sherry, petrol-like flavours (especially in the case of riesling) and earthy-tobacco notes. Taste just about any white Rioja (often aged before it's bottled) and you'll see what I'm talking about.
But to your question, finally: Yes, white wines do get old – and quickly. Most will start to taste tired two to four years after the harvest date on the label. This is especially true of light, inexpensive styles, such as Mediterranean whites. Pinot grigio, come on down. If it tastes flat and dull and has turned slightly brown (white wines get darker as they age), it has probably kicked the bucket.