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The refusal of Jehovah's Witnesses to endorse or allow blood transfusion by members is well-known and requires no recap here. Therefore we will move directly to those cites within Scripture which they use to justify the doctrine.
I have selected for the first edition of this essay three citations noted by Ron Rhodes in Reasoning from the Scriptures with the Jehovah's Witnesses [377ff]; more analysis shall be added as needed.
Genesis 9:4 But flesh with the life thereof, which is the blood thereof, shall ye not eat.
Lev. 7:26-7 Moreover ye shall eat no manner of blood, whether it be of fowl or of beast, in any of your dwellings. Whatsoever soul it be that eateth any manner of blood, even that soul shall be cut off from his people.(cf. Lev 17:11-12; Deut. 12:23-5)The
Acts 15:28-9 For it seemed good to the Holy Ghost, and to us, to lay upon you no greater burden than these necessary things; That ye abstain from meats offered to idols, and from blood, and from things strangled, and from fornication: from which if ye keep yourselves, ye shall do well. Fare ye well.(cf. Acts 21:25)
Rhodes notes  that JWs object to transfusion of blood on the grounds that it is the same as intravenous feeding. Therefore, it is argued that blood transfusion is forbidden by these passages. A comparison is noted to one who is told to abstain from alcohol, who would not therefore be allowed to take alcohol intravenously.
Is the connection from "eating" to transfusion valid?
Obviously this would not find support from OT text, long before transfusion existed; the word for "eat" ('akal) is used exclusively of consumption through the mouth (Gen. 2:16 And the LORD God commanded the man, saying, Of every tree of the garden thou mayest freely eat...).
The JW tract How Can Blood Save Your Life? responds that the medical use of blood was known in ancient times as a prescription for leprosy, epilepsy, and other ailments, and that patristic writers like Tertullian refused to even eat blood in the course of "normal" meals.
This is interesting, but hardly relevant. The "medical" use of blood via drinking would fit the definition of 'akal, and Tertullian may simply be wrong as the JWs are.
Some support seems to be taken from the use of the term "feeding" to describe intravenous feeding, but the term is used specifically because it is food products that are delivered, and if this point has any merit, then blood transfusion is never referred to as "feeding".
The question will inevitably revolve, therefore, around the reason for the prohibition, which is alluded to briefly in Lev. 17:11: "For the life of the flesh is in the blood: and I have given it to you upon the altar to make an atonement for your souls: for it is the blood that maketh an atonement for the soul."
Rhodes alludes briefly to the practices of pagan nations surrounding the Israelites that ate blood as part of the worship of false gods, or to obtain supernatural power. This is indeed the case. Consumption of blood was associated among the pagans with magical rites; in this light the prohibition on eating blood was of the same nature as the prohibition on trimming beards. To eat blood was to strive to be like what you ate, and to eat it as the seat of life was to strive to be like God, the giver of life.
Confirmation of this point is found in the instructions given for disposal of the blood of sacirificial animals: It was to be poured out on the ground (Deut. 12:16). This symbolically returned the blood to the earth from which the creature was created, by God. The disposal of the blood was associated directly with the ending of the creature's life. Transfusion is not performed in this context, and is therefore hardly to be taken into consideration.
Our cite from the NT indicates a similar background issue, as I have noted in my article on Paul: Acts vs. Epistles. Witherington [Acts commentary, 461n] notes that the decree of Acts 15 is better understood when we realize that only the wealthy ate meat with any regularity. Otherwise, a working-class Gentile usually only ate meat at public celebrations at pagan temples. Thus the prohibition on things that are strangled relates to a pagan belief that strangulation of the sacrificial animal transferred the spiritual vitality of the offering to the idol itself, and the prohibition on blood relates to the pagan practice of tasting of the blood of the sacrifice.
The decree, therefore, is comprehended best as a prohibition of attending pagan feasts and all that entailed.
A reader with a science background adds: "If blood is used as 'food', which is all the Bible prohibited and you give reasons for this, it is digested and loses all its "blood" nature. The only use the body makes of it are the amino acids and iron from the complete breakdown of the hemoglobin, etc. But in blood transfusions, the body uses the blood *as blood*, so it is not "eaten" in either a physiological or medical sense."
Summary conclusion: The acceptance of blood in the Bible is forbidden on the grounds that it involves participation in elements of pagan worship and practice. Applying this to medical transfusion of blood, where no religious element is involved, is an illicit hermenuetical step.
One might conceive of these rules covering any sort of odd religious rite that used transfusion in the same way that the ancient pagan religions used blood-eating, but for medical purposes, there is no block to blood transfusion in the Bible at all.