This is a bit of a loaded question and a very controversial topic. No other wine topic generates as much confusion and misinformation as this. The warning about sulfites on the wine bottle is required because a very small percentage of the population has a deadly allergy to sulfites. These individuals are highly allergic to raisins and many other food items that naturally contain sulfites. For others, sulfites cause no problems.
Leftover wine can be saved. The key point here is to keep oxygen away from the wine. When wine oxydizes, it degrades quickly and can soon turn into a good salad dressing vinegar.
Find a small container that will hold the wine that is left over to the point were the container is virtually overflowing. Cap the container with a cork or plug so that some of the wine does spill out. (A 375 ml wine bottle works well.) This way, you will have NO air bubble in the container.
Store the container in your refrigerator. When you are ready to drink it again, remove the container and let it warm-up to the desired drinking temperature, depending on whether it is red or white wine. You can store your wine this way for about 5 to 7 days. But I would not store it much longer than this.
Another idea for cooks is to freeze leftover wine in ice cube trays, then transfer to air-tight freezer bags to use in sauces, etc.
Most wine, is meant to be consumed within a year or two of release. Sure some of these will get better in another year of two, and if you happen let that happen; fine. Most of the worlds wines (at least 90%) are meant to be consumed young. Of the hundreds of questions we get here at the wine board a large number come from people who have hung onto wine too long. We get very few from those who popped it too soon.
White wines are not generally cellared for long periods of time although there are exceptions such as the very best Graves and Sauternes for instance.
Red wines can be and are cellared for longer periods of time. Some of the very finest reds can be cellared for several decades. A lot depends on the type of grape and the vintage. See the basics on Wine Cellaring and Aging. And we also recommend this very short video covering wine storage basics for various types of wine.
As a general rule of thumb:
Red wine, 18 ℃
White & rosé wine, 13 ℃
Champagne & other bubbly, 7 ℃
To get to those temperatures reds can normally just be kept in a cool cellar or closet. This is also called "cellar temperature." Whites and rosés can be put in the fridge for a few hours, and the bubblies longer.
It is actually easier to use an ice bucket though. Fill the bucket up with ice about 4/5ths, cover the ice with water. If the "room" temperature of your reds are over 65 degrees, immerse them for five minutes; whites and rosés for ten minutes; and bubblies for fifteen to twenty minutes. (Light reds such as Bardolino, Valpolicella, Nouveau and plain ole Beaujolais, and others of that weight should soak nearly as long as the whites.)