Once Jax finishes Tri or even gets halfway through, it's over. Jax can E to dodge half the damage from your autos, and the poison damage isn't nearly enough to get him low. His DPS is much higher than Teemo's, while simultaneously being tankier. Jax just needs to position properly so that each time Teemo harasses him, he gets a bunch of minion autos to the face. Also, a Jax who builds a doran's blade or flask and rushes a vamp scepter has enough sustain to shrug off Teemo's harass.
And Teemo, being a somewhat one-dimensional champion, does not have many options to try and outplay Jax other than some shroom positioning. If a silver Jax (me) can beat a platinum Teemo (after 6), that should be enough proof that Jax is capable of winning the matchup after a certain period of time.