This is not the sort of smile you have after indulging in your favorite chocolate bar. It's more like the kind of smile a person has when (s)he is upset, but doesn't want to upset you, too. So that person may just grin and bear it and hide his true feelings. But that kind of smile isn't helping matters in Dunbar's poem. In fact, it's making things worse because that smile is hiding the full extent of the emotional conflict he's referring to.
Line 1: Although it's "grinning," it's also lying. This sort of smile doesn't get much phonier. And by saying "grin" instead of "smile" here, the speaker also hints at the sinister quality of it, when we consider the connotations of the word "grin." Think of the Cheshire cat.
Lines 4, 10: It's also used to emphasize the duality of the emotional conflict the speaker is addressing. Notice that both lines have on one side: the "smile," and on the other: words like "cries" and "torn hearts."
Line 12: Although the speaker is talking about "singing" here, we get the sense that he's referring to the same sort of smiling appearance. Singing and smiling are usually happy things but here they are covering up the painful truth.