What is the ruling on Iraqis dealing with Christian organizations that offer aid on the condition of concealing the truth about their income, and does this constitute a blemish on the religion?

If the donor entities stipulate certain conditions for providing assistance, then it is forbidden to take the money unless the required conditions are met. Whoever takes money unjustly must return it or spend it on the needy and charitable causes, because taking money unjustly falls under lying, deception, and consuming people's wealth by false means, which are forbidden matters except in a case of necessity, which is assessed according to its extent. In a case of extreme necessity, lying is permissible to remove harm.