ANSWER: Let us look at the proof text used by one major denomination to justify their teaching that donating blood, or having a transfusion, is not God's will. It is found in Acts 15, where the apostle James is addressing a large group of church brethren (known as the Jerusalem Conference) gathered to discuss and debate whether circumcision was necessary for salvation.
James states to those in Jerusalem regarding Gentile converts, ". . . my judgment is that we do not trouble those of the Gentiles who have turned to God; But that we write to them to abstain from pollutions of idols . . . and from what has been strangled and from blood (Acts 15:19 - 20, HBFV). Basing church doctrine on a single scripture, especially in the case of donating blood which could save the life of others, is not a wise thing to do.
Note that the above verses are not discussing the donating of a bodily fluid or receiving it from another person. These verses are referring to a sometimes common practice among Gentiles (non-Jews) of EATING or DRINKING blood. It was a practice that God forbid the Israelites, who had just left pagan Egypt, from doing when he said, "It shall be a statute forever for your generations throughout all your dwellings that you eat neither fat nor blood" (Leviticus 3:17).
Notice that out of four items referenced in Acts 15:20 that three concern EATING - "pollutions of idols" (meats sacrificed to idols, see 1Corinthians 10:18 - 28), things "strangled" and "blood." God's word says to abstain from eating any of them. Even to this day some people and whole cultures injest various forms of the fluid, be it fried up, drunk, or left in the meat of an animal for "flavor." The judgment rendered in Jerusalem concerns what parts of ANIMALS humans should not eat and pass through the digestive system. It does not concern receiving blood from another human being through veins or arteries.
Transfusion is a matter which the Bible does not specifically mention. In those days, the medical technology was not available to perform such procedures. This medical practice is not any more of a sin than having an operation, a root canal performed on one of your teeth, and so on. When they were first performed, directly from person to person, they saved many lives on the battlefield.
Donating blood is also a relatively new advance of medicine that was unknown to the ancients. Although God's word is silent on this exact topic, here is a principle to consider if your are thinking about giving a donation. Hearing that Jesus had refuted the arguments of the Sadducees, the Pharisees got together to have their try at tripping him up. One Pharisee tested him with a question regarding which is God's greatest commandment. Jesus responded by stating the greatest commandments were to love him with all your heart and soul and to love others like you love yourself (see Matthew 22:36 - 40).
If your neighbor's life could be saved by donating your blood, or a transfusion of such fluid, does that not abide within what Jesus called the second greatest commandment? This certainly seems to be the case.