There's no need to burn a card after the first run because the players are all-in and have no other decisions to make. The point of the burn card is to protect against marked cards.
Cards turned in the first run are not shuffled back into the deck. There is no point because it doesn't change the equity. This may seem counter-intuitive but if you do the math you will see that it works out like that.
Here's an example. Obviously not a general proof, but hopefully this illustration suffices. Say you are headsup on the turn and all-in, needing a one-outer to win and you run it twice. There are 8 cards out (2 per player and 4 on the board).
- Win first, lose second (1/44 * 43/43) = 0.022727
- Lose first, win second (43/44 * 1/43) = 0.022727
- Lose first, lose second (43/44 * 42/43) = 0.95455
In the cases where you win, you only win half the pot, so your equity is 0.5 * (0.022727 + 0.022727) = 0.022727 = 1/44.
- Win first, win second (1/44 * 1/44) = 0.0005165
- Win first, lose second (1/44 * 43/44) = 0.0222107
- Lose first, win second (43/44 * 1/44) = 0.0222107
- Lose first, lose second (43/44 * 43/44) = 0.95506
Your equity is 1 * (0.0005165) + 0.5 * (0.0222107 + 0.0222107) = 0.022727 = 1/44.