This will kill them eventually, but the process could take months. It's possible for pupae to stay in a dormant (quiescent) state for 5 months. Insects are notoriously difficult to suffocate. Even when submerged in water fleas don't drown for 24 hours.
In this regard, can fleas survive in the water?
Fleas have this ability to survive underwater. By dissolving the grease bubble you allow water to get to the flea and they drown. Fleas are also very soft bodied. They DO have an exoskeleton that allows them protection, but soaps dissolve this and then they cannot breathe.
Can fleas die in the washing machine?
Flea infestations in the home are incredibly difficult to eradicate, but washing household items on the hot cycle, followed by a spin in the dryer is an effective method of control. Fleas will not survive a trip through the washing machine; they're survivors, but they're not that strong.