The workers have a right to demand better pay by choosing to not work there. If it's not sustainable to live in an expensive city as a burger flipper, they can leave the city and live somewhere where such a living is viable. This decreases the supply of burger flippers, meaning the market pushes the wage of burger flippers up on its own, with no intervention necessary.
Now, if the minimum wage was deemed to be too low in a place that is cheap to live in, then you would have a point, and the minimum wage probably should be increased. But if you're having trouble getting by in LA/NYC, and you refuse to move somewhere cheaper, while it sounds harsh you only have yourself to blame. The entire reason those places are expensive to live in to begin with is supply and demand. If you aren't willing (or able) to pay the price of living there, don't. It's economically inefficient to have someone who can't afford it to live in an expensive place. You may think that having more burger flippers and thus cheaper burgers in LA would be good because it would decrease the cost of eating, but it gets outweighed by the fact that the increased demand for housing increases your rent, and thus your cost of living.
Now, even if the scenario were to play out exactly as you say (less burger restaurants), then there's less places to employ burger flippers, and thus some of them will become unemployed. Thus, the actual positive effect on the minimum wage workers is lower than expected, e.g. a 25% increase in minimum wage might cause 10% of them to become unemployed. Their overall wage will go up somewhat (less than the actual wage increase), but it will be concentrated among a lower number of people. That's not a very effective solution to this problem.
It's ironic that you bring up Walmart, because as much as people like to vilify Walmart, it's one of the places that lets people get by cheaply. Without cheap places like Walmart, the cost of living would go up, so there would be even more people whose wage isn't enough to live on.