101. Is it necessary to keep the Sabbath?

Yes, it is necessary.

It is one of the Ten Commandments and is "a perpetual covenant."
Remember the sabbath day, to keep it holy. Exodus 20:8

My sabbaths ye shall keep: for it is a sign between me and you throughout your generations ... for a perpetual covenant. Exodus 31:13-16

Ye shall ... keep my sabbaths: I am the LORD your God. Leviticus 19:3, 30

Six days shall work be done: but the seventh day is the sabbath of rest, an holy convocation; ye shall do no work therein: it is the sabbath of the LORD. Leviticus 23:3

Keep the sabbath day to sanctify it, as the Lord thy God hath commanded thee. Deuteronomy 5:12

Blessed is the man ... that keepeth the sabbath. Isaiah 56:2

Whoever "defiles" it "shall surely be put to death."
Ye shall keep the sabbath therefore; for it is holy unto you: every one that defileth it shall surely be put to death. Exodus 31:14

Six days shall work be done, but on the seventh day there shall be to you a holy day, a sabbath of rest to the Lord: whosoever doeth work therein shall be put to death. Exodus 35:2

And they found a man that gathered sticks upon the Sabbath day .... And all the congregation brought him without the camp and stoned him with stones, and he died; as the Lord commanded Moses. Numbers 15:32-36

Although the New Testament is generally soft on the Sabbath, there were a few exceptions:

The author of Hebrews says that God rested on the seventh day, so believers should too.

God did rest the seventh day from all his works. ... There remaineth therefore a rest to the people of God. Hebrews 4:4-9

And Jesus told his followers to pray that the end of the world doesn't come on a Sabbath (since on that day they couldn't run away from God's wrath).

But pray ye that your flight be not in the winter, neither on the sabbath day. Matthew 24:20

No, it doesn't matter at all.

Accoriding to Isaiah, God changed his mind about the Sabbath. It is now "iniquity."
The new moons and sabbaths, the calling of assemblies, I cannot away with; it is iniquity. Isaiah 1:13
Jesus and his disciples didn't bother much with the Sabbath, and they were criticized for it.
At that time Jesus went on the sabbath day through the corn; and his disciples were an hungred, and began to pluck the ears of corn and to eat. But when the Pharisees saw it, they said unto him, Behold, thy disciples do that which is not lawful to do upon the sabbath day. Matthew 12:1-2, Mark 2:23-24

Jesus saith unto him, Rise, take up thy bed, and walk. And immediately the man was made whole, and took up his bed, and walked: and on the same day was the sabbath. The Jews therefore said unto him that was cured, It is the sabbath day: it is not lawful for thee to carry thy bed. John 5:8-10

And therefore did the Jews persecute Jesus, and sought to slay him, because he had done these things on the sabbath day. John 5:16

If a man on the sabbath day receive circumcision, that the law of Moses should not be broken; are ye angry at me, because I have made a man every whit whole on the sabbath day? John 7:23

And the ruler of the synagogue answered with indignation, because that Jesus had healed on the sabbath day, and said unto the people, There are six days in which men ought to work: in them therefore come and be healed, and not on the sabbath day. Luke 13:14

And Jesus encouraged others to break the Sabbath, saying that the Sabbath was made for man, not man for the Sabbath, and that he was Lord of the Sabbath. (So he and his followers are free to break it.)
And Jesus answering spake unto the lawyers and Pharisees, saying, Is it lawful to heal on the sabbath day? And they held their peace. And he took him, and healed him, and let him go; And answered them, saying, Which of you shall have an ass or an ox fallen into a pit, and will not straightway pull him out on the sabbath day? Luke 14:3-5

He said unto them, The sabbath was made for man, and not man for the sabbath: Therefore the Son of man is Lord also of the sabbath. Mark 2:27-28

Paul* told his followers to observe the Sabbath on any day they liked. It's not important and no one should be judged for how they keep (or don't keep) the Sabbath.
One man esteemeth one day above another; another esteemeth every day alike. Let every man be fully persuaded in his own mind. Romans 14:5

Let no man therefore judge you in meat and drink, or in respect of a holy day, or of the new moon; or of the sabbath days. Colossians 2:16

* Colossians was probably not written by Paul, but was forged by one of his followers. Bart Ehrman, The New Testament: A Historical Introduction to the Early Christian Writings, 3rd ed. (2004), chapter 23.

Christian Responses
Grace Communion International
The author of the SAB answered
Annotated Skeptic's Annotated Bible: Our fathers did eat manna in the desert (Exod 16)