Does Overzealous Fan stop "when I attack" units effects from resolving when he uses his ability?

No — the attack trigger still resolves; Overzealous Fan doesn't cancel it.

The "When I attack" ability was already placed on the chain when the unit gained the Attacker designation (Rule 383.4.e), and the Fan's defend trigger resolves before it (defender triggers go on top, LIFO). Moving the attacker to its base does not stop its trigger from resolving.

What changes is whether the effect can actually do anything: if the trigger references "here" or otherwise needs the unit to still be at that battlefield — e.g. "deal damage to an enemy unit here" — it will fizzle, because "here" is checked on execution and now refers to the unit's base (Rule 359.3.f.2, the Yasuo/Fight or Flight example). Effects that don't depend on location or combat status (e.g. "when I attack, draw 1") resolve normally. If the attack trigger targeted the Fan specifically, that instruction is also ignored since the Fan is in the trash.