As the previous exploit poster said, you can finish your random dungeon dailies without being online. The trick is making it so you can't be kicked from the group. I'm rewriting this post in order to explain how to do it step by step since the previous exploit poster just alt+f4'd out.
1) Get saved to the instance ID. Basically just kill the first boss with the group.
2) Die. You have to die in the instance...so pull a trash mob or get aggro and let yourself die. In some cases, you can just stay dead and you'll be unable to be removed from the group...although this isn't 100% true every time.
3) Once you are dead, you can alt+f4 and go about your day. Come back later, and if the people in the instance were able to complete it by 4 manning it, you'll get credit for the whole instance once you log in.
If the people were not good enough to finish the heroic, it's a fail for everyone, so I wouldn't attempt doing this as the tank or healer unless you know that someone else in the group can use dual-spec to finish.
+ it's kind of a dick move....6):
Tested and worked in Nexus and HoL.
Also, not an exploit more of a tip that I thought i'd mention...a way to earn some free abyss crystals or dream shards is right after killing the final boss in any random heroic or random dungeon... type /afk and just sit there without rolling on the items from the last boss. Wait for the roll to almost expire, and the ppl who ran the instance with you will get tired of waiting and most likely leave the instance. If not all of them, at least a few will leave, and you'll have a much better chance of winning whatever.