Well there is the obvious relationship in that they both deal with Joseph Kony and the Lords Resistance Army and its aftermath in Uganda.
Sam Childers, the subject of Machine Gun Preacher is known to the Group Invisible Children as seen in this Google cache from Invisible Children's blog where IC says:
Actually, let’s start with a disclaimer: Invisible Children does not in any way endorse this man’s actions or methods.
Now that that’s settled, we can say ‘what the….?’ In brief, this American preacher is leading a militia of Sudanese and Ugandans with the sole aim of killing Joseph Kony. He claims that this is his God-ordained mission.
There is another distinction that Invisible Children and Sam Childers share; they may be more hype than hope.
According to the Pittsburgh Tribune Sam Childers' orphanage has problems:
Some neighbors around Nimule want Childers removed and the orphanage turned over to local leaders. While Childers was on an American publicity tour, Nimule community leaders told a reporter for the magazine Christianity Today that he dishonored his agreement with them. The resulting story detailed a local government inspection that found the orphanage with no food, little medicine and shelters on the verge of collapse.
Similar charges have been levied against Invisible Children. In a recent post from journalist Michael Wilkerson he notes that:
(Invisible Children)... it spends too much money on administration and filmmaking, while still touting its on the ground NGO-style projects. Also, apparently it's never been externally audited.
Claims that less than one-third of the money raised actually goes to help children in Uganda have been made.
A recent showing in of Kony 2012 in Uganda angered many people. An Al jazzera article notes that:
Towards the end of the film, the mood turned more to anger at what many people saw as a foreign, inaccurate account that belittled and commercialised their suffering, as the film promotes Kony bracelets and other fundraising merchandise, with the aim of making Kony infamous.
Will it drive up sales of the Machine Gun Preacher dvd when it comes out? Probably not. The movie itself didn't even gross $1 million. I doubt the studio is going to see a real reason to push the dvd of a movie that lost $29 million.