Irish dance tunes are generally continuous flows of melody with no rests. This presents a problem for the wind instrument player, who sooner or later is going to have to do some inhaling. You can certainly grab a breath by cutting a long note slightly shorter. But what are you going to do when the next long note is four measures away and you're on the verge of running out of gas? How are you going to avoid dropping a note?
Well, that's just it. You're going to drop a note. It's no big deal, everybody does it, and it's not cheating. On the contrary, skillful placement of breaths in the flow of a tune is one of the hallmarks of a good traditional flute player. It defines your phrasing, adding punctuation to what would otherwise be an uninterrupted stream of eighth notes.
As with pretty much everything else discussed in this site, the best way to learn how this works is to listen to good players doing it. But there are a few general guidelines that may help you figure out what you're hearing. As a rule, the second eighth note of each beat in a reel or hornpipe is the least important and may be safely dropped. Obviously you don't want to drop the note that comes on the downbeat, and the third and fourth notes can be considered as a lead-in to the next phrase. What happens if you break a tune up this way is that your phrases all begin on the upbeat, which makes your playing sound very smooth and finished.
You'll also hear a lot of players dropping the fourth note of the beat in reels. This makes your phrases begin on the downbeat, creating a foursquare, stately feeling. To me, this style of phrasing seems particularly characteristic of East Galway/Clare players.
In jigs, the first and third notes of each beat are the most important -- again, the downbeat matters the most, and the third eighth leads onto the next beat. So the second note is the best option for dropping.
Where you breathe will end up being a feature of your personal style, so it's important on the one hand to let yourself develop an instinct for it, and on the other to cultivate your instinct by listening to how other players do it.
Remember to keep your breaths short and quiet. Two well-placed breaths, no matter how close together they are, are better than one long gaspy wheeze that disrupts the tempo. Keeping your tongue low in your mouth will help to minimize breathing noise.