Does Ezreal dashing deal damage with his effect if he gets moved to base by Janna?

No, Ezreal's damage does not deal damage if Janna moves him to base before his trigger resolves.

Here's the sequence: Ezreal attacks, his "When I attack" trigger goes on the chain. In response, you play Janna, Savior with Reaction to the same battlefield. Janna's "When you play me" trigger goes on the chain above Ezreal's and resolves first (LIFO), moving Ezreal to base. When Ezreal's trigger then resolves, "here" re-evaluates to his current location — his base, not the battlefield. Since there's (presumably) no enemy unit at his base, the ability has no valid target and whiffs.

This is the same principle as the Atakhan + Janna interaction (FAQ #11312) and the general rule from FAQ #7639: moving a unit away from the battlefield causes its "here" referent to shift, making the trigger fail to find a valid target.