Usually, no. They are able to activate in rain for example, but will not work if submerged in liquid fully. Certain members of the Jedi Order, usually of species who can live underwater such as Mon Calamari or Nautolan, make a modification to their lightsabers during creation so that they can be used fully underwater by them. The only time I know of that shows this is in the original Clone Wars TV series, on Mon Calamari, when Kit Fisto jumps from the deck of his Acclamator and ignotes his lightsaber underwater.
Luke Skywalker used his original lightsaber (the blue one) underwater in Splinter of the Mind's Eye,
but whether or not that is still cannon since Disney now owns Lucasfilm is unknown.