In short: Pure Burmese do not have blue eyes as adults!
As adults, pure Burmese should have an eye colour that is a shade of gold. It is notoriously hard to get really rich eye colours in some coat colours (e.g lilac), but on the show bench, the deeper gold the eye colour, the better.
All kittens are born with blue eyes, but a pure Burmese will often start to develop the traditional eye colour as young as 4 or 5 weeks. Blue eyes on a Burmese would usually indicate that either it is a very young kitten (likely under 6 weeks), or if it is an older cat, blue eyes would indicate that it is not a pure Burmese.
By the time a kitten is 10 – 12 weeks old, their eye colour will be quite well developed, and in a pure Burmese, it should be developing towards a striking golden colour.
As mentioned above, some coat colours (such as lilac) often struggle a little to get the ideal depth of eye colour that other coat colours (such as Red and Brown) usually produce with ease. Eye colour is largely genetic. I myself own a stunning Chocolate girl with fantastic characteristics (coat, nose break, chin, temperament, etc.) but she inherited her father’s weak eye colour. Her other character characteristics still mean that she is show quality (and does quite well on the show bench), but her weak eye colour is certainly a disadvantage.
If you are specifically after a cat with blue eyes, you may like to look at a Siamese or even a Tonkanese (Burmese crossed with Siamese).
Below are some examples of some Burmese with fantastic eye colour:
Here are some examples of young kittens that have their baby blue eye colour:
Here are some kittens who are transitioning eye colour:
