I think Dany has at least shown some worrisome tendencies. She couldn't be talked out of crucifying the masters in Meereen (one for each cross along the road). Jorah had to talk her out of killing all the masters in Yunkai when they took back the city, by reminding her about the Mad King (and that he would be dead himself under that form of justice). Tyrion had to talk her out of burning both Astapor and Yunkai to the ground (IIRC), again by reminding her about the Mad King (and telling about his plan for King's Landing).
To her credit, most of the time she listens to her advisers when they council restraint instead of just going with her instincts, but for someone who claims to know what her father was, she sure does need to be reminded and have the dots connected for her each time an opportunity arises to do something it seems like he would do. Her "Burn them all!" plan being both the last and the worst yet doesn't exactly show growth in this regard.
I don't think Dany is mad now, and I don't expect the story to go the route of Dany going mad, but I think there's enough evidence to support it as a possibility at this point.