What is the technique used by and effect of the quotation, "Two of the fairest stars in all the heaven, /Having some business, do entreat her eyes / To twinkle in their spheres till they return" (2.2.15-17)?

The figurative language of this passage employs imagery, as well as the personification and metaphor explained above.
Romeo's use of celestial imagery, which is any language that conjures a vision, smell, sound, feeling, and/or taste, suggests that he finds Juliet's beauty other-worldly. By linking Juliet's eyes with "two of the fairest stars in the heaven," Romeo is expressing the depth of his infatuation with Juliet, and he believes her beauty to be as deep and unknowable as the universe.
Romeo's use language that suggests that Juliet is unreachable, as the stars are, is an indication that he has already put her on a pedestal of sorts and that he idealizes her. His idealization of her is evidence of his romantic personality; holding her at a distance allows him to imagine her as perfect.

In these lines, Romeo essentially says that two of the prettiest stars from the sky had to go away for some business and have asked Juliet's eyes to shine in the sky in their stead while they are gone.  This quotation makes use of the poetic device called personification.  This is when human attributes are given to something that is not human.  Obviously, stars are not sentient, and they lack the ability to ask anyone to do anything.  However, in using this device, Shakespeare achieves a level of whimsy, as if Romeo's world were magical now with Juliet in it.  It also helps to show just how beautiful Romeo believes Juliet to be; in this sense, the line also contains a comparison, a metaphor, wherein he claims that her eyes are as bright and beautiful as the stars are.  When he ascribes the action of "twinkl[ing]" to her eyes, he compares them to stars because it is stars that twinkle, not eyes.