The only known workaround at the moment is to only have 0 or 1 quests stored. If you have 2 you won't get a 3rd one and if you have 3 it might delete one.
I know, some of you guys really like to improve the odds of getting better quests by having 2 quests stored when getting a new one, but for now it's the only way to get a new one. Hopefully Blizzard is working hard to fix this bug asap.
It seems that there is a bug with the third questslot atm.
I had same Problem a few days ago. I didn´t get a new quest, cause I had two slots full. That day I made all my quests and then I got a new one next day. Today I also got a new one. Now I have two slots full. I suspect if I wouldn´t do any quest today, I won´t get a new one tomorrow.
My friend had same problem. Two slots full. But he didn´t do his quests and he didn´t get a new quest.
I just can give you the advice to do your quests every day and just let one at the most till Blizz will solve the problem. Also write a ticket or in forum of Hearthstone to let them know.
As far as I know they just give you advice to do some regular steps which normally help you out. These are thinks like reseting passwort, change language, try logout and login again, aso. But it doesn´t seem to help anybody, because it ist a generally problem with the third questslot and not a Problem some less people have. Really hope they will fix it soon.
Completing all my quests seemed to resolve this issue for me--at least, for the time being. I did, and started receiving new daily quests the next day--no sign of the missed legendary quests, though. :-(