If you're not so fast an overzealous fan, does the fan still die from defending?

Yes, if you use Not So Fast to counter Overzealous Fan's triggered ability, the Overzealous Fan still dies.

According to RiftJudge FAQ #8702 (which reflects the current Unleashed rules update), the "kill me" instruction in Overzealous Fan's text is a cost within instructions that must be paid upfront to finalize the ability onto the chain.

Sequence of Events

1. Trigger Condition Met: Overzealous Fan gains the Defender designation and the ability triggers.
2. Payment and Finalization: To place the ability on the chain, its controller must pay the cost by killing the Overzealous Fan. The fan is now in the trash.
3. Ability Added to Chain: With the cost paid, the triggered ability is finalized and placed on the chain as a pending item.
4. Reaction Phase: The opponent (or the active player) may now play Not So Fast to target and counter that triggered ability.
5. Resolution: Not So Fast resolves, countering the Overzealous Fan's ability. The ability is cleared from the chain and its effect (moving an attacking unit) does not happen.

Key Clarification

Because the "kill me" requirement is a mandatory cost paid to put the ability onto the chain, the Overzealous Fan is already dead by the time your opponent has the opportunity to play Not So Fast. Countering the ability prevents the effect (moving the attacking unit) from occurring, but it does not "bring back" or save the Overzealous Fan that was already killed to pay the cost.