I only started doing this recently. I needed to get about 8 rivets redone over the course of a year. The third time I came back the guy asked me what I do with my skates after I play since he knows I play a ton.
I now take my footbeds out, pull my tongues all the way forward and hang them upside down. They're so much dryer when I have to play the next day and my rivets still look brand new inside the boot.
I also use Superfeet. I picked up a lace puller to get under the heel portion of the insole in order to pull it out. No excuses, guys!
I don't mean to be preachy, but this goes along with airing/drying your stuff out. We all know how expensive this sport can be, so take the extra minute to pop your insoles out and air out your skates (as well as the rest of your stuff) properly. You will get so much more mileage out of your equipment and avoid the stench that the rest of your teammates have to put up with, but you somehow find humorous.
Take them out, keeps the rivets from rusting for sure and was told to do so by the pro shop I bought the skates from. I don't do the boot dryer though, just place them in a closet filled with dehumidifiers.