To be eligible for the additional rewards you must solo queue for a random level-85 Heroic in the role that is currently being Called to Arms, and complete the dungeon by killing the final boss. Every time you hit these requirements (there is no daily limit) you'll receive a goodie bag that will contain some gold, a chance at a rare gem, a chance at a flask/elixir (determined by spec), a good chance of receiving a non-combat pet (including cross faction pets), and a very rare chance at receiving a mount.
By the way, I disagree with the premise that the current queue times for dps are crazy. They are long, but they aren't crazy long. So long as you have something else to do (quest, gather, dailies, ...), the queue times are just right to give you a pleasant alternation between two kinds of activity.
Nonetheless, shorter queues would be better. What can be done?
I suspect Call to Arms won't make much of a difference. Already, tanks can queue instantly, so they have a very strong enticement to queue as a tank instead of as a healer or dps. The additional benefit of getting gold out of it is swamped by the benefits they already have.
A better angle, though one much more difficult to implement, is to make it less onerous to tank. This can be hard to do, because tanking is much more demanding than the other roles.
One possible solution would be to lower the gear requirements for tanking in heroics. Make the mobs hit less hard, so that characters can queue for heroics with less fear of wiping the group. As the game is balanced right now, tanks that meet the minimum ilvl requirement are fairly squishy. Some might say this is making the game too easy on tanks, but if we are seeing tanks not sign up or heroics, the game must already be fairly hard for them. A nice thing about this solution is that it only affects heroics. As far as I know there is no shortage of tanks for raids.
Another possibility would be to make tanking rotations even simpler, or at least, make a simpler rotation be sufficient. I find that when tanking with a new class, there's a tanking equivalent to grid lock where you get so focused on your threat rotation that you lose track of where all the mobs are. If tanks had a couple of fewer buttons to worry about, then they'd find it easier to focus on all the other things tanks do. This strategy, granted, can easily backfire. Many complained in Wrath that tanking many fights was terribly boring, and that's why Cataclysm tanks have so much more to do.
A third possibility, and not something they can implement immediately, is to reduce the ratio of tanks that are needed for a group. Why not have 8-man dungeons with one tank and one healer?
Overall, it's a tricky problem. As usual, it's gratifying to see that Blizzard is thinking about these game design problems instead of just raking in their dough. I don't think Call to Arms will solve the tanking shortage in Dungeon Finder, but it's a simple thing that is easy to try.
UPDATE: Jar on Rank 4 Healing Touch takes up the same question.