I've got more than one battle.net account, but only one has Heroes of the Storm closed beta access. Now that I've downloaded the game, can I play with either account, or will it not work with my alt?
It will only work with whichever account has the closed beta access. You won't be able to play with your alt account.
The game will show up on both battle.net clients, but you will be able to play with the one which has the access. On other account, you'll get authentication error.