You can make a really good argument that the Edmonton Oilers are the Calgary Flames' most bitter rival, but probably the team that the Flames have most frequently had important games against is the Vancouver Canucks.
Examples? The Flames have faced the Canucks seven times in the playoffs, twice heading to the Stanley Cup Final after beating the Canucks in the first round (the Canucks have gone to the finals a few times after beating the Flames, too), and they've faced them more times in the regular season than any other NHL club. Oh, and they also had a pretty sweet line brawl back in 2014 that seemed to kick-start the rebuild.