I have never heard a market-oriented economist argue that a rise in the minimum wage boosts the demand for labor. You might try this argument: "The government is certifying that these workers are worth this much. The government is defining the market price. Entrepreneurs will believe that price and hire workers in the expectation of finding an equivalent or even superior marginal product. The government said that was the right price."
No go. Market-oriented economists instead claim that entrepreneurs "see through" to the real marginal products of these laborers. The demand for labor, rather than rising, would fall and unemployment would result.
So what happens when the Fed "sets" short-term interest rates or influences other prices? What is postulated by monetary misperceptions theories, including Austrian business cycle theory? Entrepreneurs no longer see through to the fundamentals. Instead, entrepreneurs are taken to believe this Fed-influenced rate is the correct price and they make their plans accordingly.
What is the difference between these two cases?
Several commenters at MR are on the right track I think, but let me take a shot at it with some different language.
The difference is that when central banks create excess supplies of money and push interest rates below their natural level, there really are resources available to borrowers. Yes, the interest rate is sending a false signal about underlying time preferences, but the loans banks are making at the lower rate really do represent resources to the borrower. Of course those resources are not the result of real savings elsewhere and represent losses to the rest of society, but they are still a gain to the borrower. Hence there's nothing to "see through" in the sense that there's no binding constraint on the borrower in the same way that the underlying productivity of the worker binds the employer who is considering hiring at the minimum wage.
In supply and demand terms, a minimum wage law drives a wedge between quantity supplied and demanded with a price floor. Inflation pushes down the interest rate not by directly imposing a price but by shifting the supply of loanable funds curve outward through forced savings in a way that creates a credit market equilibrium interest rate (the market rate) that no longer reflects underlying time preferences (the natural rate). The fact that we trade time in the form of money, rather than somehow directly, is what makes this divergence possible and what distinguishes it from the labor market. Such a separation is not possible there and price controls will create shortage and surpluses.
As several commenters note: the minimum wage is a price control, but an inflation-driven fall in the market rate is a quantity adjustment masquerading as a false price. The low interest rate is the result of the underlying problem: too much credit. The credit, though, is "real" to the borrowers, even if it is an illusion with respect to voluntary savings.
During the boom, borrowers really do acquire resources and some will profit handsomely if they get out in time. The fact that the errors embedded in the boom take time to manifest, along with the idea that inflation is ultimately a distortion on the quantity axis not the price axis, enables borrowers to not face the immediate constraint that they do in the minimum wage case.
Tyler seems to be forgetting that money is different. Its "loose joint" properties mean things don't work quite the same way there.