A solution would be to reward those that play non DPS classes, while not punishing the DPS players. I think the best solution would be increase the rewards for Tanks and Healers. An example could be:

1. Tanks and Healers can earn rare mounts when queueing (solo) via the Duty Finder.
2. Tanks and Healers can earn more Tomestones every week then the DPS players.
3. Tanks and Healers can will be rewarded with bags which has different rewards, like minions, housing items and bardings.
4. Tanks and Healers will earn alot more Gil when doing content via the Duty Finder.

Tanks and Healers will be rewarded for queueing and DPS will be rewarded by having reduced waiting time in the queue. The DPS players that get jealous and want to earn those rewards aswell, they will play a Tank or a Healer and inturn get rewarded, while helping other DPS players enter the dungeon much faster.

Its a WIN/WIN.

These options will reward the Tank and Healer, while not punishing the DPS. If a DPS wants to have these rewards, then they need to play those two roles. If you are playing a DPS, then you need to deal with the bad parts of that role, which in MMORPG is longer queue then the other roles.