"The boy in the blue shirt" is correct. "The boy in blue" is also correct, but it doesn't indicate just his shirt. This would mean that the majority of his clothing is blue. "The boy in blue shirt" is incorrect; it needs the "the" before "blue," just like your first example.

The boy in the blue shirt - this is fine. The boy in blue - this is fine. He's wearing blue clothes. When you say this, you need to say "the" blue shirt. A shirt is a countable object. You do not need "the" with my second example, because the color is not a countable object. You could also say "The boy in the blue" because it is "understood" that you left the piece of clothing off......"shirt", but this is not so normal.