Would adding a small amount of dish soap to a sprinkler system help prevent run-off by acting as a surfactant?

Yes, it works exceptionally well if the lawn is very dry. And it works for the very reason you cite. When outdoor pots dry out, it makes re-wetting them much easier as well.

Not really and Dish soap will burn your grass some. Shaklee used to advertise Basic H, essentially dish soap, as an insecticide. Does work great, will drop a grasshopper in mid hop when sprayed. Also supposedly will help plants grow, but too much burned mine.