Assuming it could be done at all, which it can't. The article says "If you could do an impossible thing you'd get a strange result", which, duh.
But also, you could have a Zeno's Paradox of particles - one with half the energy, one with 1/4th the energy, one with 1/8th...
That's not quite as ridiculous as it sounds: a microwave photon could be centimeters long, and a lower-wavelength photon could be meters or more. You could think of the shutter as an LCD screen, which could change from clear to dark in less than the time it would take for a meters-long photon to pass through it.
You can't actually do it, though the speed of light isn't the reason. The calculation is for an infinitely thin, infinitely black barrier. You could do an approximation of it, but the complexity of the approximation swamps the already-complex wave form the theory gives.