What Happens when Passenger Refuses to Fly Boeing 737 Max?

What recourse will a passenger who is scheduled to fly commercial on that model of plane have if they refuse to board because of safety concerns?

None. The airline and the relevant regulators are the competent authorities to determine what types of planes are safe to fly, not the passengers.

Of course, in many cases, airlines will do things such as rebooking or refunding tickets that they're not obliged to do, because it's good PR. However, I would expect that the collateral "Airline admits passenger who refused to fly on one of their planes might have a valid point" would be worse than the PR of "Airline won't help passenger who refused to fly on the plane that was provided."

As of 12th March, the BBC reports reports that Southwest is allowing customers booked on a 737 MAX-8 to change their reservation, but not offering refunds. However, this is just Southwest's standard policy of not charging fees for rebooking, which applies to all their flights.

And, as of 13th March, Boeing has grounded all 737 MAXes. From this point, the question is somewhat moot. Passengers whose flights are cancelled or rescheduled as a result of this have the same rights as they would for any other cancellation/rescheduling.

If a regulator has ordered that aircraft grounded and an airline continues to fly it, contact the regulator. Beyond that, you would have a hard time arguing that the fears were "well founded". They may not be baseless, but still not "well founded".

At that point, it would be a passenger voluntarily refusing to board. No recourse.

Simple, No recourse. You may always politely ask for rescheduling to different equipment. "It doesn't cost anything to be nice" -- Bear Bryant

NPR.org reports that:

The Federal Aviation Administration says it plans to require a series of design changes to the Boeing 737 MAX fleet after a pair of fatal plane crashes including one yesterday in Ethiopia that killed all 157 people on board.

Boeing confirms it's in the process of enhancing the flight control software for the plane, including the MCAS software.

CBC.ca reported that:

The Air Canada Pilots Association, the union for Air Canada and Air Canada Rouge pilots, sent out a statement urging Transport Canada "to take proactive action to ensure the safety of the Canadian travelling public."

Not exactly clear what they meant by that but the Transport Minister (of Canada) would consider it "premature" to ground the 41 planes of that model owned by Canadian airline companies according to CBC.

I'm afraid that doesn't provide further options for the consumer. So I guess all that's left is for people who think the precautionary principle requires them to not fly in such planes would be to cancel their bookings, try to get a refund from the seller and purchase new tickets trying to ascertain what plane they'll be flying on. If as some people have stated it's possible to guarantee that you'll fly in the plane described when booking, then you may choose a direct flight with an airline which doesn't operate the said plane...

If an airline has scheduled a flight using a particular aircraft, then it means they deemed it safe enough to fly. Acknowledging that a passenger who refuses to board has "well-founded fears" is basically admitting that they are knowingly putting other passengers and the crew in danger. So no, that won't happen: either they cancel the entire flight, or you will have to refuse to board.

The recourse you have as a customer (whenever or not you end up boarding a 737 Max) is to take note of the airline whose safety standards are lax enough to keep flying that model while everyone else grounds it, and avoid using its services in the future, with optional public shaming in social media.

In many places around the world the point has become moot because the entire EU has suspended the Boeing 737 Max flights. Even before that Australia, Austria, Belgium, France, Germany, Iceland, Ireland, Italy, Malaysia, the Netherlands, Oman, Singapore, Turkey and the United Kingdom have announced suspending of the flights. The UK, Oman, Singapore, Australia, Ireland and France and Norwegian Airlines suspended the whole Boeing 737 Max range while India, Dubai, China, Indonesia, Malaysia, Iceland, Germany and the airlines LOT Polish, TUI Airways, GOL Linhas Aereas, Aeromexico, Aerolíneas Argentinas, Cayman Airways, Comair Airways, Eastar Jet, Jet Airways, Mongolian Airlines, China Airlines, China Eastern, China Southern, Lion Air and Silkair have suspended the Max 8 model. Turkey suspended Max 8 and 9 models (CNN.com, US and Canada are the only two nations still flying many Boeing 737 Max planes). A different source adds Royal Air Maroc to the bunch and includes individual reports for each.

Because some of the decisions bar a variant of the plane from entering the airspace, the effect is far ranging despite some countries/airlines still flying the plane; which therefore will have to be domestically, or from non-ban countries/zone to non-ban countries/zone and outside the airspace where the model is barred from flying, for instance from the continental U.S. to Hawaii or to Canada surely still works for the time being...