Is trinity force viable on champions that doesn't use mana?
I've been seeing people who use shen, rengar, zed, etc. use trinity force.
It seems to work but since they don't use mana, is getting trinity force viable?

Yes, it is very solid on Shen and Lee Sin. You only lose a few hundred gold for the mana cost, but it is hardly inefficient.

yes, the raw stats aren't the only things you want to look at, the passives (movement speed, slows, and the sheen extra damage on auto-attack after an ability is used). the sheen alone makes it great imo for ad casters (aside for riven, whose combos go faster than sheen CD, and her own cooldowns are too long afterwards)

Triforce only gives 250mana which is worth 500g. That isn't much of its overall value which is lost.

Triforce is mostly an offensive tool with most of its value going into offense and utility. You want a champ that will likely only get triforce for offense with maybe a lategame atma or whatever. You want one that ideally needs the Snare effect but doesnt REALLLY need the snare (as they will want Frozen mallet instead). A champ that can SPAM spells and has a high base AD is the best bet. You generally dont want autoattackers to get triforce since they prefer AS/crit/AD by itself.

Mundo can use Triforce effectively, once he has some decent defenses set. A mundo might go with a Warmog for health and regen/sustain, tabi and wits end and spirit visage for armor/MR, an Aegis for more defense maybe or chain to atma endgame, and a triforce to end up with. The triforce can quickly be triggered due to his dmg booster and mallets and will do a lot of extra damage. The movespeed and snare are both very nice to have, though not absolutely essential due to his cleaver slow.

Triforce on shen is a bit odd since shen is a Tank. You dont build 4100g offense items on a tank in most cases. Mundo is a damage soak/tanky DPS so needs to balance offense with defense.

lee Sin is a physical caster and all his stuff scales off of AD. As such triforce is very odd for him. He will want bloodthirsters that give great sustain and recovery, plus power his big AD ratios. Ditto with a pantheon, Jayce, etc. TF can work but its not ideal.