As Cantilena said, NEVER EVER try to remove tarnish yourself. Using chemicals or TOOTHPASTE is impossibly stupid - you can wash silverware in the sink, but you certainly cannot do that to a flute - and even the most microscopic amount of abrasive will find its way into the mechanism. And those sites where any idiot can post cleaning advice - yeah, I'd really trust MY instruments to them!!
Once you flute comes home from the shop, keep a 3M anti-tarnish strip in the case - you need to change it out every so often. With the case closed, it absorbs free sulfur, and your flute will stay non-tarnished longer. However, you STILL need to wipe it down after you play - EVERY time! I use a microfiber cloth - not the kind with loops, which catches on the rod pins - but the velvety kind used to clean computer screens. I am wiping down a ten thousand dollar flute every day, so am VERY careful - and it stays looking good. This is only ONE of the instruments I own - and make my living with - so they ALL have to be in great shape, and I do not risk anything homemade in their maintenance. Sometimes, I leave it out on a flute stand, and then after a day or so, I see the beginnings of tarnish - reminds me that "I'll be right back" does NOT always happen. IN THE CASE, except when you are playing.
Professional flutist, certified teacher since 1971, competition judge in NY state. I currently own over $100,00 worth of fine instruments - I do not risk them for ANY reason.