I'd have to say it doesn't quite work for teemo as much as it might work for singed.
Not to mention is a HUGE nerf to singed as it forces him to leave lane for more poison. This would mean you want his Q to ONLY have the resource of poison while his mana pool suffers.
Singed would have to be buffed to compensate for this nerf... out of all his moves which one do you think Riot would buff?
here's a hint:** THEY'D BUFF HIS Q**
His Q damage can already get really high but with this buff/nerf you'd change singed(and teemo by extention) to a point at which
having a finite amount of something = justifying it being OP in some way/shape/form
I'm not even gonna talk about how Teemo would be buffed because I already explained how singed's Q the ONE MOVE he would be using for this "poison" meter would become overpowered due to damage buffs.
-up to this point I've assumed you meant poison would be going along with mana similar to Jhin or Annie's passive bar the next few sentences talks about if it replaces mana entirely-
Singed builds mana to become tanky... so if you take away his mana then his passive will need to be changed Which means more work for riot on the coding end.
Teemo needs his high mana costs to balance him out.
Without mana costs teemo's shrooms which are already annoying would be intolerable.
Not to mention singed and teemo would become inherently stronger the more towers were lost thanks to base being closer.
Overall the poison resource wouldn't work for singed and Teemo.
Maybe if a new champion was made out of a poisonous gas or was like "Muk" from pokemon as a toxic living gelatin.
The finite "poison resource" bar just not worth it to make.