Penguin's card says
compare this to the Belt:
Am I right in thinking I can reroll the roll for the Penguin Resource? Do I need LOF to the character who used it this turn to do so?

No. The character may be using the effects, but it's not actually assigned the Resource. The Resource is assigned to your entire force, which is not anything you can draw LOF to.

You can't reroll the Penguin Resource roll.

Chris there are some things that can allow you to re-roll it though. Any power that allows you to force a re-roll without drawing a line of fire would work. For instance Ancient One and his power:

THE ORIGINAL SORCERER SUPREME: Once during your opponent's turn, Ancient One may force that opponent to reroll any die roll, ignoring the original roll.

You could use this to force an opponent to re-roll their penguin resource roll. I know corner cases suck, but for this one I thought it prudent to point it out.

Well I was mostly referring to Probability Control, but you're right that there may be exceptions.

This one isn't so bad though. Corner cases don't bother me as much as mass general shenanigans.

Would I be correct in assuming the effects of Penguin only last until the beginning of your next turn?

THE PENGUIN ALWAYS TAKES HIS CUT: The Penguin is a resource assigned to your force and is not placed on the map. Once, at the beginning of your turn, you may choose a friendly character. That character can use the power indicated on the dial's current click and modifies its combat values as per the current click. At the end of your turn, if the chosen character was given a non-free action this turn, roll a d6 and compare:
1 = Remove The Penguin from the game and heal the chosen character 1 click.
2-5 = Turn the Penguin's dial to the right once. If The Penguin is KO'd it is removed from the game.
6 = Turn The Penguin to any click on the resource dial.

Correct...