How would the poem "We Wear the Mask" be different if Paul Laurence Dunbar had written it in more informal language or without as much structure?

This is a great question that really could open up to a number of interpretations.
"The mask" mentioned in the poem is a facade Dunbar says is "worn" by African Americans in society. This "mask" portrays one set of socially acceptable emotions and behaviors, regardless of what a person may actually be feeling.
The formal, elevated diction of the poem reflects this idea. Although the African Americans who wear "the mask" likely speak in a less formal dialect common in Dunbar's other works, that way of speaking is not "respectable" or allowed in White society. Much like these people must wear the mask, they must also speak in a way that plays up their "respectability" and hides their own cultural practices and way of speaking. Therefore, the diction of the poem reinforces its theme in a way that a more informal diction could not.

Paul Laurence Dunbar wrote much of his work in an African-American English dialect of the Antebellum South and also, on occasion, employed the Midwestern regional dialect favored by his friend and patron James Whitcomb Riley. The elevated diction and formal structure of "We Wear the Mask" is very different from either. The language of the poem is tragic—with "torn and bleeding hearts," "tears and sighs," and "tortured souls." Christ is even invoked as the witness of suffering.
Dunbar never says who "we" are, which means the question is open to interpretation, but in the context of the poet's life and work, it is overwhelmingly likely that he is referring to African Americans and their position in American society. The primary reason for his use of this structure and language, therefore, might be to give dignity and even grandeur to the struggles of African American people.
There were all too many detractors willing to cast aspersions on the intelligence and intellectual attainments of African Americans in the late-nineteenth- and early-twentieth century, which was when Dunbar was writing. Many were incredulous that a black man could aspire to a career in literature at all. To have written in-dialect would have served to reinforce their prejudices, and Dunbar wants to give such people no excuse to write him off as he articulates the tragic position of his race.

This is a great question. The key argument to be made here is probably that the structure of the poem itself can be equated with the "mask" which the speaker says we wear at all times. A formal structure like the one Dunbar chooses here makes use of traditionally accepted elements, such as a regular rhyme scheme and the use of archaic language, like "nay" and "O." This immediately casts the poem into a form we recognize, because we have seen many poems structured and pitched in this way before. As such, the structure not only lends cohesion to the poem, it also confines it.
How Dunbar's poem might have looked without the "mask" of such a formal poetic structure and use of language, we cannot say. However, it is very probable that the way Dunbar writes here is not the way he would have expressed his message in spoken language. As such, he is not taking off the mask in protesting about it, but rather using the "mask" of traditional poetic form to indicate how confined we are by that mask.