There was a time, in the 1800s and on into the early 1900s, when a good dress shirt was not just a shirt. It was part of a carefully arranged ensemblée. The collar, for example, would be a separate piece, as would be the cuffs. There might also be a marcella bib or placket to hide the shirt studs (only a heathen would use buttons!) And just as you would starch the collar and the cuffs, you would starch the shirt. A boiled shirt is specifically a white dress shirt boiled in highly starched water, resulting in a very stiff carapace, much stiffer than what you can get with a cold or lukewarm wash. The resulting shirt was stiff as a board. There was a good reason why a gentleman was never wrinkled: his shirt simply would not allow it.

The process of boiling, drying, and then ironing a fully starched shirt is arduous, and it must be repeated every time the shirt is washed. Understandably, the boiled shirt has gone the way of the petticoat. What with a tie and collar at the top, and a waistcoat at the bottom, and a placket in front, even a lightly starched shirt is not liable to wrinkle excessively in the course of a formal evening. And if you happen to be doing the sort of strenuous activities made popular by Fred Astaire and James Bond, you are likely to find a stiff shirt too much of a hindrance to your movement. The upshot of this is that it is nearly impossible to find a place to either buy or launder a boiled shirt unless you live in the UK -- and even there they can be difficult to find, and quite expensive.

Nowadays, if you see an article of clothing that claims to be boiled (most often a sweater, but occasionally a shirt), it was most likely boiled in plain old water in order to make the fabric softer and more comfortable, and will not require any fancy upkeep. Primarily because it isn't fancy.

Further reading
Yes, people still wear these
How To Boil A Shirt