When you play Void Seeker on a unit on the battlefield and an enemy flashes the chosen unit back, will the spell still do damage and resolve?

No, the spell will not deal damage, but it will still resolve.

According to the RiftJudge FAQ (ID 1794, 5984, 3134, 6470), when a unit targeted by Void Seeker is moved to a base (or any non-battlefield location) in response to the spell, the unit becomes an invalid target for the damage portion of the effect.

Here is the sequence of events:

1. Chain State: Void Seeker is played, targeting a unit at a battlefield.
2. Response: The opponent plays Flash (or a similar effect) to move the targeted unit to their base.
3. Resolution (LIFO): The Flash effect resolves first, moving the unit to the base.
4. Void Seeker Resolution: When Void Seeker attempts to resolve, it checks for its target. Because the unit is no longer "at a battlefield," the damage portion of the spell cannot be executed and is ignored (per Rule 359.3.e.9 and 359.3.e.10).
5. Remaining Effect: Because Void Seeker contains two separate effects separated by a period ("Deal 4 to a unit at a battlefield. Draw 1."), the draw effect is not conditional on the damage succeeding. Following the "do as much as you can" rule (Rule 359.3.e.11), you will still draw 1 card.