Minstrel shows were ostensibly a caricature of African-American culture, reaching the height of their popularity in the mid-19th Century.
By and large, rap and hip-hop developed in the 1970's as a natural progression from disco, jazz, beat poetry, raggae, and other African Diasporic cultural genres interacting with each other and creating something new.
Humor, as an entertainment genre, created and performed by African-Americans doesn't have much of a presence in American popular culture until the 70's.
In this way, there's not much of a direct relationship between minstrel shows and contemporary African-American entertainment in terms of "lineage." However, racism - of which minstrel shows are one example - is a frequent topic of discussion.
Example - In the "When Keeping It Real Goes Wrong" sketch of Chappelle's Show Season 2 episode 7: Dave plays Vernon Franklin, an African-American yuppie who feels like an "Uncle Tom" because he likes and gets along with his white co-workers. So instead of letting an awkward comment from co-worker and mentor Frank Murphy slide, he over-reacts.
Frank: Vernon, great job buddy. You da man! Give me some skin, huh?
Vernon: Get your m*f*in hand out of my face, man. You heard me m*f*er get your hand out of my face! What you think this is - just shake my hand like a man! .. 'Gimme some five, on the black-hand side, with all this crazy jive'? That's b*s*. Do you want a little soft-shoe? Should I juggle some watermelon? Man; f* all that!
Obviously, the humor is in the over-reaction. But at the same time, Chappelle is tapping into the sensitivity many people have concerning racial stereotypes: The comment about "dancing a soft-shoe" is a direct reference to minstrel show caricaturing of African-Americans.