Can you use Throne Room to Mine a Copper to a Gold?
Yes, you can trash Copper -> Silver -> Gold, since Mine requires you trash cards from your hand, and then gain them back into your hand, setting you up for another play.

Copper -> Platinum with a King's Courted Mine is also possible (and incredibly devious!)

For clarity, Mine reads:


  Trash a Treasure card from your hand. Gain a Treasure card costing up to 3 Coins more; put it into your hand.


And Throne Room reads:


  Choose an Action card in your hand. Play it twice.


So the order is (with a hand like Throne Room, Mine, Copper, X, X):


Play Throne Room
Choose Mine
Trash Copper from hand; Gain Silver into your hand
Use Mine again
Trash Silver from hand; Gain Gold into hand
Profit