There are no visible indicators for fatigue level in PS:T. Dak'kon and Annah are the only characters affected by it at all.

If I'm not mistaken, the way this works is that the characters gain one "point" of fatigue every 4 in-game hours. Once that value reaches 6 (i.e. they've gone without sleep for 24 hours straight), they start complaining about being tired. If you ignore that and keep them awake, they will also start getting penalty to Luck (again, 1 point every 4 in-game hours - I think you can actually see the effect if you enable on-screen attack roll messages). Since Luck reduction ultimately makes them useless in combat (and, in Annah's case, thieving), this will eventually get problematic.

The item you're referring to is the Heart Charm - it temporarely sets Fatigue to 0, but doesn't actually "cure" it, so it'll just keep stacking up until your party rests. (And spellcasters, including Dak'kon, need to sleep to memorize spells either way.)

Assuming your party can sleep regularly, pretty much the only point in the game where exhausted party members can become a problem is Sigil exploration beyond the Hive+underground (because worldmap travel takes time). There are ways to avoid this, but since there are no time limits, you can always rest somewhere before continuing on the quests.