Sulfites are a natural byproduct of wine, and a part of many things we consume, like dried apricots, tortillas, blue cheese and molasses. Sometimes winemakers choose to add additional sulfites to the trace ones that naturally occur, as a way to help prevent wine from spoiling.
Unfortunately, about 1 percent of the population is sensitive to sulfites (this number goes up to 5 percent of asthma sufferers) and my heart goes out to them. Sulfite reactions sound scary, from wheezing to asthma-like attacks, and sometimes more severe reactions can include digestive issues, dizziness and difficulty swallowing. If you think you have a sulfite allergy, please get tested.
Because of how severe sulfite allergies can be, the U.S. government thought it would be helpful to require that wines with sulfites in excess of 10 parts per million get labeled with the "contains sulfites" disclaimer you've no doubt seen. Other countries don't have that same labeling requirement, which does create some confusion as the same wine can carry the warning or not, depending on where it's sold.
So, for medical reasons real and imagined, people seek out wines that don't have the "contains sulfites" warning. Organic wines have "no added sulfites," but they may still contain trace amounts of naturally-occurring sulfites.
So can you remove sulfites yourself? Coffee filters won't do it, but there are products out there that use a specific type of polymer filter that can supposedly filter out some of the sulfites. But are you filtering enough of the sulfites out so you won't have a reaction? Are you stripping the wine of any other flavors? You can also add a few drops of hydrogen peroxide, which will oxidize the sulfites and neutralize them. But does the hydrogen peroxide change the flavor of the wine? You really should consult with your physician to make sure you're making the best choices for your health.
For people who don't have a medical reason to avoid sulfites, please don't blame sulfites for your hangovers or headaches. I worry that "sulfite-free" is the wine equivalent of "gluten free"—sounds healthy, but is only medically necessary for a select group of people.