Can dish soap really be used to kill ticks and fleas?
First part of the answer - Yes fleas (Siphonaptera) can be drowned. But not as easily as the internet would lead you to believe.

There are many claims on the internet (and printed works) expounding on how simple it is to drown fleas.  The best science I found so far on the topic is in Forensic Entomology: An Introduction By Dorothy Gennard; John Wiley &amp; Sons, Apr 30, 2013; section 4.3 which reflects a 1985 work by Simpson K. (Journal of the New York Entomological Society 76: 253-265, not finding this online). Gennard is using the fleas present on a human body at death to determine how long the body had been submerged.  Findings - 


A flea submerged for up to 12 hours will appear to dead, but can revive in about 60 minutes after being removed from the water
A flea submerged for 18 - 20 hours will appear to dead, but can revive in  4- 5 hours after being removed from the water
It takes 24 hours of full submersion to fatally drown a flea 


Side note; lice (Phthiraptera) can be fatally drowned in about 12 hours.



Second part of the answer - Yes soap can facilitate the death of fleas in bathing

So far the best I have is Medical and Veterinary Entomology by Gary R. Mullen, Lance A. Durden Academic Press, Apr 22, 2009 


This work suggests the process of washing the flea (and the pet) removes integumental waxes on the fleas body and they die from desiccation (dehydration).


More Research

I have been unable to find solid research specific to fleas (Siphonaptera) and soap (surfactants), so I had to reach in to the general insect works for answers.  There are a number of pesticides which include surfactants in their make up to increase effectiveness but these seem to be to an aid in the delivery of the pesticide not because the surfactant has any additional killing power.


  a surfactant may affect the efficacy of an insecticide by its influence on wetting, spreading and run-off rather than by its influence on cuticular penetration


The Physiology of Insecta, Volume 6 edited by Morris Rockstein

There is some science suggesting soaps can help break down cell membranes but they seem to be most effective on soft bodied insects (which fleas are not)


  Soaps... kill insects by disrupting the exoskeleton and breaking down cell membranes.  Soaps generally work best against small soft bodied insects such as aphids, scale crawlers, meatybugs, and young caterpillars as well as spider mites.


Destructive Turfgrass Insects: Biology, Diagnosis, and Control By Daniel A. Potter

Summary

There are multiple online reports of "drowning fleas" and killing them with soapy water.  After looking for the science there does not seem to be much support for these fatal claims. There is little doubt that giving your pet a soapy bath in the tub will help remove the fleas, they may even appear dead.  But in all likelihood these fleas are just waiting to dry out and be revived, hopefully in the city sewer.  If you bath your pet in the yard, and than let the dry pet in the yard again the next day the clean pet and clean revived fleas will likely reunite. Soapy water is more likely to facilitate removal of the comatose flea from the pet and carry it down the drain. So it is reasonable to have noticeable decrease in the flea population after a bath.  But excessive bathing has it's own risks and given the life cycle of a flea soapy baths alone are unlikely to be a final solution.