Are bounties active in Heroic Story missions?

Bounties try to be very specific about what they what you to complete, and for a good reason. It leaves it open to the player about how they want to complete it.

For example, if the bounty says "Defeat enemies on Mars using grenades." then you could be patrolling, in a strike, a nightfall, an adventure, a story mission, a weekly heroic mission, whichever. As long as you are on Mars, and not in the wrong gametype such as Crucible or Gambit, then any grenade kill will work.

Just be sure to read the requirements carefully and it should be obvious how and where to complete the bounties.