As a dramatic monologue, "The Bishop Orders His Tomb" is meant to be read as if someone is speaking out loud. We say more about this form over in "Form and Meter," but as far as the poem's sound goes, it might best be described as conversational.
That's not to say, though, that Browning abandons all sonic technique in this poem. When it comes to sound stylings, we actually get several examples of two main poetic techniques: alliteration and assonance.
The alliteration comes with the repetition of the beginning sounds in words, as in "She, men would have to be your mother once" (4). The repeated M sounds emphasize the familial connection the bishop mentions here. Later, we get D words: "What's done is done, and she is dead beside" (6). These underscore the finality of the mother's death. Finally, we get several S words that seem to hiss out the bishop's anger at Gandolf's tomb selection: "Shrewd was that snatch from out the corner South" (18).
Alliteration is not Browning's only sound game, though. He uses assonance to create sonic echoes of repeated vowel sounds. In particular, he seems to get a lot of mileage out of long E sounds. We hear it in "And thence ye may perceive the world's a dream" (9), "Swift as a weaver's shuttle fleet our years" (51), and "Shall ye contrast my frieze to come beneath?" (55). In every case, the repeated long Es create a heightened echo that lends a kind of intensity to the lines.
Throughout the poem, Browning is sneaking in subtle, but powerful sounds to lend a punch to the bishop's directions to his sons. We're meant to hear the urgency in his voice, which, of course, is totally appropriate. The man is dying, after all, and that tomb is not going to build itself.