No, the myth that pigeons mate themselves to death is not true.
The average wild pigeon lives for 6 years and during this time they will raise at least 12 clutches of eggs, typically these will all be with the same mate as pigeons are monogamous.
If a pigeon’s mate dies then the remaining pigeon will then look for a new mate, this is a process that can take a few months or even longer (females take longer to choose a mate than males).
Given that the average pigeon is a parent to over 30 pigeons over the course of its life it is simply not true that pigeons die as a result of mating.
There are quite a few animals that this is true of but pigeons are not one of them.
Sure, pigeons do die after mating as most creatures do, however this is certainly not as a direct result of having mated.