Did a bit of research on this because I was curious. There isn't much in the way of credible information on how playoff payments work, but here's what I've been able to gather:
Teams that make the post-season pay into a league-central playoff fund. The exact percentage of revenues they pay is based on that team's total gate revenues (top 10 pay 50%, middle 10 pay 40%, last 10 pay 30%). Any money left in other funds will be included in the total payouts as well. Out of those funds, teams are paid based on how far they progress, and receive the same amount whether they play 4 or 7 games. The amount they receive pays for players, coaches, trainers, and other staff, so comparatively, it doesn't come to much. There are also bonuses for certain milestones and awards (such as the President's, Art Ross, Conn Smythe, and Norris trophies, and of course, the Stanley Cup).
That info's based on the CBA, and the only credible article I could find from 1998:http://www.washingtonpost.com/wp-srv/sports/capitals/longterm/1998/stanleycup/articles/know16.htm