Does Teemo Strategist have to activate his ability if Irelia moves into the battlefield?

Yes, Teemo, Strategist’s ability is mandatory, and you must place it on the chain if the conditions are met.

According to his card text: "When I defend, choose an enemy unit here and reveal the top 5 cards of your Main Deck..."

The Ruling:

• Mandatory Trigger: Teemo’s ability does not contain the word "may," meaning it is not optional. When he is designated as a defender in combat, the ability triggers automatically (Rule 383).
• When it happens: If Irelia moves into a battlefield where Teemo is present, Teemo gains the defender designation (assuming combat is active). Because his triggered ability lacks a "you may" clause, you are required to place the ability on the chain.
• Resolution: When the ability resolves, you must choose an enemy unit at that battlefield (in this case, Irelia) to be the target of the damage effect. You cannot choose to "not activate" the ability to avoid choosing a target or dealing damage.

Important Considerations:

• Deflect: Because Irelia has the Deflect keyword, your opponent must pay [A] to choose her with an ability. Even though Teemo's ability is mandatory for you to place on the chain, Irelia's Deflect protection applies to being "chosen" by abilities. Therefore, to resolve Teemo's effect targeting Irelia, your opponent would have to pay the Deflect cost if they wish to protect her, or the choice would be considered invalid/illegal depending on if they choose to pay the tax.
• Timing: As per RiftJudge FAQ #10534, this ability triggers and resolves before combat damage is calculated or applied. If the damage from Teemo’s ability kills Irelia, she will be removed from combat before she has the chance to deal any combat damage.