Get Religion has a nice summary, replete with links, of the state of the issue. I can see how one would feel obliged to help those in need, but condoning law-breaking? Stated another way, the duty to be a Good Samaritan certainly justifies groups that aid any who present themselves for assistance, but it doesnt yield a stance on immigration law. Nor, for that matter, does an argument regarding "the dignity of the human person" deprive a country of the right to decide who may and may not become a citizen.
Those who profess concern about the plight of people entering the U.S. illegally are obligated both to provide assistance to those who need it and to obey the law. Its entirely possible to assist someone with his or her immediate material needs while also informing the authorities, who can then make their own determinations regarding the persons status. I realize that those who wish to remain in the country illegally will thus be less likely to seek assistance, but churches and religious organizations cant be in the position of actually encouraging illegal behavior.
Its also, I think, appropriate for those who say they care about illegal immigrants to devote a greater portion of their efforts to addressing the conditions that prompted their immigration in the first place. There are at least three things they can do. First, they can assist American citizens in finding and qualifying for jobs that would otherwise be filled by illegal immigrants. To the extent that the demand dries up, theres less incentive to cross the border illegally. And to the extent that there are needy Americans who could work, they should have the first crack at the jobs. Second, religious groups could put pressure on businesses not to hire illegal aliens. While this would likely raise the cost of doing business, and raise the costs to consumers, so be it. Why not conduct a campaign promoting goods and services made in America by American citizens? Some folks at least are willing to pay more for things produced under conditions that promote social justice; this is just another one of those. Third, they can expand their humanitarian mission work in Latin America, helping to ameliorate the economic, social, and political conditions that prompt people to leave their homes.
Or is it easier just to bash the "hard-hearted" government or to impugn the religious motives of your opponents?