What is the contrast John Donne is making between "sublunary lovers' love" and the "refined" or heavenly love between the speaker and the audience?

In order to nail this question, you need to look at the stanza in which he actually uses the phrase. 

Dull sublunary lovers' love 

   (Whose soul is sense) cannot admit 

Absence, because it doth remove 

   Those things which elemented it. 

(Donne 13-16)

You don't have to go any further to get your answer started. These "dull, sublunary lovers" (which literally means sub-lunary, or below the moon, or terrestrial—these lovers are earthly, whereas Donne and his...

In order to nail this question, you need to look at the stanza in which he actually uses the phrase. 

Dull sublunary lovers' love 

   (Whose soul is sense) cannot admit 

Absence, because it doth remove 

   Those things which elemented it. 

(Donne 13-16)

You don't have to go any further to get your answer started. These "dull, sublunary lovers" (which literally means sub-lunary, or below the moon, or terrestrial—these lovers are earthly, whereas Donne and his wife know a heavenly sort of love) cannot allow a lover to leave because their love is entirely based around the other person's presence. This is in an age long before Skype or FaceTime, so a trip or any separation between two lovers could mean months or even years without seeing the other person. These "sublunary" lovers cannot stand that. Donne and his wife, as implied by the contrast he sets up, can.

He says as much in the next stanza, when he says that he and his wife "Care less, eyes, lips, hands to miss" (Donne 20). This nonphysical love means that they love each other on an otherworldly, or spiritual, level. When Donne says "Our two souls therefore, which are one," he means it (Donne 21). Their love is not about bodies. It is about souls. Where those "dull, sublunary lovers" cannot bear to have their lover leave because they only know how to love a person's physical being, Donne and his wife can be apart for years because they love each other's souls. 

He ends the poem with a conceit about their souls acting like the type of compass one would use to draw a circle. The farther apart they are separated, the larger they grow, because they are still connected. When they are together they may not be as vast, but they are "erect" and strong and hold each other up. (Since this is Donne, be sure not to miss the fact that the word "erect" can be a sexual pun for "erection." Donne loved writing about sex.)

Basically, those "dull, sublunary lovers" only know how to love on a physical level. It takes lovers who understand each other on a divine level, like Donne and his wife, to truly experience love on a spiritual level. By doing so, one manages to grow during separation rather than split apart.

NOTE: I mention Donne and his wife because for this poem, it is academically accepted that Donne is writing this to his wife. So the audience that the question asks for is Donne's wife, and the speaker is Donne himself.