ANSWER: The Bible has NEVER clearly commanded vegetarianism and eating meat is certainly permitted in New Testament times. We know that Jesus, when He appeared to his disciples after His resurrection, ate fish (Luke 24:41 - 43).
We know that under the law of Moses an enormous number of animals were ritualistically slaughtered and offered to God by various burnt sacrifices (See Leviticus 1, 3-7). The priests were allowed to eat the meat from what they offered under certain limitations (Leviticus 7:31 - 35). So it cannot be a sin to kill an animal intrinsically, for God had the priests kill thousands of animals over the centuries for these animal sacrifices. They represented the sacrifice of Jesus for humanity's sins in advance symbolically. The most direct case of this comes from the Passover. Each Israelite family (maybe with the neighbors included) was to take a lamb or goat, kill it, and (in the first Passover before Egypt) put its blood on the door posts and lintel overhead the threshold of the door. They were to eat it by morning, and burn the leftovers before daybreak. This is described in Exodus 12. Many centuries later, John the Baptist proclaimed that Jesus was the lamb of God offered to take away the world's sins (John 1:29, 36).
Abraham and Noah also had to kill animals in order to make sacrifices to God, which was LONG BEFORE the time of Moses and the establishment of the Levitical priesthood (Genesis 8:20; 22:7 - 8, 13). Under the law of clean and unclean foods, Israelites were allowed to eat cows, sheep, and goats, but weren't allowed to eat pigs, camels, and rabbits (Numbers 11, Deuteronomy 14:3 - 21). These general rules are the greatest proof God DOES NOT require vegetarianism of his people.
Although God didn't like how Israel in the wilderness demanded flesh to eat when they had a bad attitude, and punished them for it, He was still willing to provide them with a huge number of quail to eat in one special situation (Numbers 11:4, 13, 31 - 33). Obviously, it cannot be said that eating them was deliberately sinful, or God wouldn't have sent them the quail.
There is one interesting counter argument worth examining. The argument is that Adam and Eve, when they were created, were not directly told they could eat the flesh of animals. Although this may be true, the Bible DOES state that God gave man authority over the animals and was told to rule over them (Genesis 1:28 - 30). There is also no doubt that God, after Noah's flood, gave humanity permission to eat meat (after the blood is drained away - see Genesis 9:3 - 4).
We know that animals will not eat each other during the millennium (e.g. lions will not eat calves, see Isaiah 11:6-9). But since there will animal sacrifices in the millennium (Ezekiel 40:39; 42:13; 45:15-25), after Jesus' return, we cannot say those restrictions will apply to people also.
Even though God has given us authority over animals and allow us to eat them we should not ABUSE them. Balaam was criticized by his (miraculously) talking donkey and by the Angel (Messenger) for beating his donkey when he (the animal) saw the Angel (Messenger) of God with a drawn sword in front of him and would not move (Numbers 22:22-35).
In conclusion, the Bible does teach that the eating meat is allowed (Deuteronomy 12:20).