Teemo Strategist triggers versus Irellia Fervent?

Yes — Teemo's "When I defend" trigger fires and is mandatory when Irelia moves in, but it comes with a cost and doesn't buff her.

• When Irelia moves to his battlefield, Teemo gains Defender and his trigger goes on the Chain automatically (no "may"), resolving before combat damage (FAQ 10815).
• Deflect applies: to choose Irelia with the trigger, Teemo's controller must pay [A] as a mandatory additional cost. If they can't or won't pay and Irelia is the only enemy unit there, the trigger can't finalize a legal choice and is removed from the Chain (Rule 402.4).
• Irelia's own "When you choose or ready me" does not fire — "you" refers to her controller, and Teemo's trigger is controlled by her opponent.
• If the damage kills Irelia, she's removed from combat before dealing combat damage.