Sceadeau clued me in yesterday that there's a feature in Hearthstone to swap out one of the daily quests if you don't want to do it. I feel a little silly for not knowing about such a feature for so long, especially since it turns out to be pretty simple, but now I know. And other people should know too!
Basically each day you get assigned a quest at random. You're allowed to store up 3 quests at a time so you don't need to play every day to keep on top of your quests but they also don't just stack up forever. Quests can be for a variety of tasks and can be worth either 40, 60, or 100 gold. I've always just done them when they show up, or shortly thereafter.
It turns out, however, that each quest in your log has an X on it in the corner. I assumed this was to delete the quest in case you hit your limit of 3 and wanted to get a different one the next day. That's not quite how it works. Instead it deletes the quest and immediately gives you a different one. It also removes all the Xs from all your quests for the rest of the day.
So it would seem that each day if you get a 40 gold quest you should send it back. If it comes back as a 40 gold quest then you should just leave it sitting there and send it back the next day, too. (Days change at midnight PST, so 4am out here.) And then only worry about doing a 40 gold quest if you've got a big backlog of 40 gold quests.
Realistically I'm not going to go through all that effort. If I have a 40 gold quest for casting spells I'm not going to play games without playing spells all day. I'm just going to get my 40 gold and move on with my life. But it does mean that any time I get a quest I will be sending it back for a chance at a bigger gold one. And if I get a quest to win twice with a class I don't like then I can just ignore it and reroll it again the next day too instead of forcing myself to draft warrior.