What is the relationship between Barbara Kingsolver's The Poisonwood Bible and colonialism texts such as Things Fall Apart and Heart of Darkness?

The Poisonwood Bible has much in common with such critiques of colonialism, but it also moves beyond them, incorporating some realities from the postcolonial age. 
This is partly because it is based on the author's own experience. As she says in her Author's Note, 

I was the fortunate child of medical and public-health workers, whose compassion and curiosity led them to the Congo. They brought me to a place of wonders, taught me to pay attention, and set me early on a path of exploring the great, shifting territory between righteousness and what's right.

The Poisonwood Bible was exhaustively researched. In addition to travel and personal interviews, the author drew on many fiction and nonfiction works about colonial and postcolonial Africa, including Things Fall Apart. Her novel comes with a bibliography. It builds on the colonialism texts.
What The Poisonwood Bible has in common with the two novels you mention is that it is a critique of colonialism that nevertheless recognizes complexity. In all three novels, the colonized culture is not portrayed as perfect, but at the same time the destructiveness of colonialism is clearly seen. Another theme in all three novels is the sheer volume of cultural knowledge required to even begin to interact with an African culture, and how the outsiders coming in can't help but spend their days in mistaken assumptions, misunderstanding, and confusion. In Heart of Darkness, this confusion is portrayed from the colonizer's point of view. In Things Fall Apart, it is shown from the point of view of the local people. In The Poisonwood Bible, we start out with the outsider's perspective, but we slowly gain more of an insider perspective as the Price girls learn more about the culture. And ultimately Leah moves into a dual insider/outsider perspective as she crosses cultures, marries Anatole, has his children, and raises her family in Zaire.
This crossing of cultures is the main thing that sets The Poisonwood Bible apart from the other two books. Leah Price makes a serious effort to "go native." Her experience shows the degree to which culture-crossing is possible, as well as the degree to which it is not possible. And through her thoughts, we get to see quite a bit of her motivations for choosing to live as she does.
Basically, in Leah we have a character which is very rare in colonial times but very common in postcolonial times: a repentant colonialist. She comes to Africa believing every word her father says, and wanting to help him "bend Africa to his will" and do "God's will" in the Congo. During the year the family spends there, her eyes are opened both to the beauty and intricacy of the local culture, and the huge flaws in her father's character. She is still strongly motivated to do right, but now she identifies with the Congo and wants to do right by the people of Kilanga, especially Anatole. Her transformation coincides with the revolution in the Congo.
With her perceptiveness and her conflicted feelings, Leah brings us a postcolonial phenomenon: White guilt. In one scene, she beats herself up for the complete drag that her family has been on the village. Anatole replies,

"Don't try to make life a mathematics problem with yourself in the center and everything coming out equal.  ... No, you shouldn't [have come here]. But you are here, so yes, you should be here. There are more words in the world than no and yes."

Leah, confronting for the first time the damage that her family and her race are doing, flips from thinking she is the hero to thinking that she is the villain. Anatole tries to help her see that life is more complex than that. This is a lesson we could all stand to learn.
Though Leah crosses cultures, she never completely gets over white guilt. She continues to feel responsible for the behavior of America and for how she is perceived. She continues to take very personally the frustrating ups and downs of sociopolitical life in Zaire. In a sense, she spends her life trying to atone for being white.
This full-blown case of white guilt (rather than just having a few doubts) is both a major theme in The Poisonwood Bible, and a modern/postcolonial rather than a colonial phenomenon. It is also a theme that appears in some of Barbara Kingsolver's other books.