What happens if you die trying to beat the Thunder Dragon’s Challenge

You need to talk to the professor outside the upper exit of the academy, then head to the spot marked Plaza on the map, which should trigger a cutscene.

There are more steps to it after that, but I'm not sure if you want more than that or not.

Here's the changes that I was told by characters in the game or have observed so far:

• The slot in the quest selection screen will have the goddess insignia on it.

• You can Skip almost all cinematic and dialog sequences

• Keep all bugs and treasures from previous adventure

• Some characters will acknowledge that you're playing in Hero Mode as they speak with you** What they say to you will change telling you what has changed or allowing you to skip some explanations.

• You lose hearts twice as fast as you normally do

• All hints are unlocked at Sheikah Stones

• Maximum strength Skyward Strike always
  Every Skyward Strike you perform will have the strength as if you had used the Master Sword, including the faster charge. After getting the actual Master Sword, charging is instantaneous.

• Hearts and Heart Flowers no longer appear normally in the game*
  Monsters, pots and other sources do not drop hearts anymore. You will have to heal by using potions, fairies, sleeping or sitting in chairs. *Equipping the Heart Medal will provide a reduced chance for Hearts and Heart Flowers of appearing.

• Fi will remember how many of each type of monster you defeated from your previous adventure
  When Fi analyzes a targeted enemy, the count will continue from where you left off.

• Your high score at Bamboo Island and other places that keep records will remain from the previous adventure

• The Boss Rush challenge now includes the game's final battle, in addition to the other eleven battles you could access in regular mode. You must have defeated the boss at least once before they are playable.

• Beating the game this time around will allow you to save just before the final battle and you get a Triforce icon in the quest selection screen.

To all hardcore Zelda fans out there, the game is a lot more challenging in this mode so it's definitely worth it to play through.