No, not necessarily. The traps and leaves have a natural lifespan. Each time a trap closes, it must grow to open again, and eventually it has grown the maximum amount it is genetically capable or programmed to grow, at which time it doesn't close anymore. But it still can live, stay green and provide food to the plant through photosynthesis for quite a few months, until the trap and the leaf die of old age, replaced by lots of new leaves and traps.
Sometimes an insect is so large that part of it stays outside the trap, which can in some cases encourage fungal growth that can also attack a trap and blacken it. However, Flytraps that are grown in healthy and happy conditions often do not turn black during or after digesting an insect, but weak plants with thin, light-starved leaves can and often do turn black when fed.