Do dolphins lay eggs?

No, Dolphins are mammals. They breathe air, are warm blooded, give live birth to babies and nurse their young.

http://www.crystalinks.com/dolphin.html

No. Dolphins are mammals; they give birth to live young.

https://floridakeys.noaa.gov/animals/dolphins.html

The term dolphin refers to two distinct animals. The more familiar dolphin is a mammal and gives birth to live young. The other type of dolphin is a fish, and is also known as "dorado" or mahi-mahi (satguru). It lays eggs.
https://en.wikipedia.org/wiki/Mahi-mahi

I agree with Gnomon as I have managed to catch one off the coast of Florida here is a picture of one:

http://www.defenders.org/dolphin/basic-facts

You must remember...There is the Dolphin (mammal) and the Dolphin Fish. Two distinctly different creatures.

The term dolphin is also applied to a group of aquatic mammals (see dolphin, mammal). The fish known as dolphins are classified in the phylum Chordata, subphylum Vertebrata, class Osteichthyes, order Perciformes, family Coryphaenidae.

https://oceana.org/marine-life/ocean-fishes/common-dolphinfish