Nazarite Vow

What do they demand from Paul?
That he participate in a Nazarite vow and even pay the expenses for the Nazarite vow for four others who apparently couldn't afford it themselves, as means to demonstrating to the Jewish believers that he is a pious Jew.

What is a Nazarite vow?
The Nazarite vow is an Old Testament vow that God gave to Moses for the Jews who were especially consecrating themselves to God: "1 Then the LORD spoke to Moses, saying, 2 “Speak to the children of Israel, and say to them: ‘When either a man or woman consecrates an offering to take the vow of a Nazirite, to separate himself to the LORD, 3 he shall separate himself from wine and similar drink; he shall drink neither vinegar made from wine nor vinegar made from similar drink; neither shall he drink any grape juice, nor eat fresh grapes or raisins. 4 All the days of his separation he shall eat nothing that is produced by the grapevine, from seed to skin. 5 ‘All the days of the vow of his separation no razor shall come upon his head; until the days are fulfilled for which he separated himself to the LORD, he shall be holy. Then he shall let the locks of the hair of his head grow. 6 All the days that he separates himself to the LORD he shall not go near a dead body. 7 He shall not make himself unclean even for his father or his mother, for his brother or his sister, when they die, because his separation to God is on his head. 8 All the days of his separation he shall be holy to the LORD. 9 ‘And if anyone dies very suddenly beside him, and he defiles his consecrated head, then he shall shave his head on the day of his cleansing; on the seventh day he shall shave it. 10 Then on the eighth day he shall bring two turtledoves or two young pigeons to the priest, to the door of the tabernacle of meeting; 11 and the priest shall offer one as a sin offering and the other as a burnt offering, and make atonement for him, because he sinned in regard to the corpse; and he shall sanctify his head that same day. 12 He shall consecrate to the LORD the days of his separation, and bring a male lamb in its first year as a trespass offering; but the former days shall be lost, because his separation was defiled. 13 ‘Now this is the law of the Nazirite: When the days of his separation are fulfilled, he shall be brought to the door of the tabernacle of meeting. 14 And he shall present his offering to the LORD: one male lamb in its first year without blemish as a burnt offering, one ewe lamb in its first year without blemish as a sin offering, one ram without blemish as a peace offering, 15 a basket of unleavened bread, cakes of fine flour mixed with oil, unleavened wafers anointed with oil, and their grain offering with their drink offerings. 16 ‘Then the priest shall bring them before the LORD and offer his sin offering and his burnt offering; 17 and he shall offer the ram as a sacrifice of a peace offering to the LORD, with the basket of unleavened bread; the priest shall also offer its grain offering and its drink offering. 18 Then the Nazirite shall shave his consecrated head at the door of the tabernacle of meeting, and shall take the hair from his consecrated head and put it on the fire which is under the sacrifice of the peace offering. 19 ‘And the priest shall take the boiled shoulder of the ram, one unleavened cake from the basket, and one unleavened wafer, and put them upon the hands of the Nazirite after he has shaved his consecrated hair, 20 and the priest shall wave them as a wave offering before the LORD; they are holy for the priest, together with the breast of the wave offering and the thigh of the heave offering. After that the Nazirite may drink wine.’ 21 “This is the law of the Nazirite who vows to the LORD the offering for his separation, and besides that, whatever else his hand is able to provide; according to the vow which he takes, so he must do according to the law of his separation.” (Numbers 6:1-21)

Which element of the Nazarite vow absolutely was NOT for Christians?
The "sin/trespass" offerings: "Then on the eighth day he shall bring two turtledoves or two young pigeons to the priest, to the door of the tabernacle of meeting; and the priest shall offer one as a sin offering ... and make atonement for him ..." (Numbers 6:10-11) "... bring a male lamb in its first year as a trespass offering" ... (Numbers 6:12) "... one ewe lamb in its first year without blemish as a sin offering ..." (Numbers 6:14) "... Then the priest shall bring them before the LORD and offer his sin offering ..." (Numbers 6:16)

Why not?
The Lord had granted the sin offerings to the Israelites to point forward to Jesus' sacrifice on the cross which atoned for all sins of all Christians: "For the death that He died, He died to sin once for all ..." (Romans 6:10). Any sacrifice aimed at "atonement" (Numbers 6:11) for sins and trespasses contradicts the power, glory and finality of Jesus' sacrifice.

Were the elders of Jerusalem right or wrong to demand this Nazarite vow from Paul?
They were wrong to demand it from Paul, as well as to let it take place within their flock in Jerusalem.

What do they recommend the gentile believers (accompanying Paul) do to dispel the rumors?
Nothing more than what had been written previously: "But concerning the gentiles who believe, we have written and decided that they should observe no such thing, except that they should keep themselves from things offered to idols, from blood, from things strangled, and from sexual immorality" (Acts 21:25).

When had that been written?
Years earlier when Paul and Barnabas had come to Jerusalem to discuss the inapplicability of circumcision to gentile believers (see Silas).