JetBlue is offering the service for one reason and one reason only: as a marketing gimmick.
The $4 fare barely covers the mandatory taxes and fees. The actual revenue they're getting is about $1 or so. I highly doubt the people that bought these tickets had any need to go from Burbank to Long Beach.
But by announcing the deal, it got lots of people to look at their website, which then will make them consider JetBlue the next time they need to fly somewhere.
It's not really providing a practical transportation alternative during this period of construction.