Does Christianity or the Bible forbid polygamy and widow remarriage? I've seen people advocate widow celibate for the reason of a happy reunion of the marriage bond in Heaven. As I understand it, this appeal is based on the rejection of polygamy, which is based on the rejection of lustful feelings towards a woman outside the marriage which is considered a lechery.

According to a Czech master's thesis I've read, Christianity has surmounted polygamy as Jesus establishes the equality of man and woman: "There is neither Jew nor Gentile, neither slave nor free, nor is there male and female, for you are all one in Christ Jesus." Galatians 3:28

However, I don't consider this a sufficient basis. So what does Christianity base the rejection of polygamy and widow remmariage base upon? Thank you!
Christianity forbids polygamy but allows widow remarriage.

I agree with you that Gal 3:28 is not usually cited as support for the rejection of polygamy, because the passage context is about unity of Gentiles and Jews in Christ (notice the contrasts of social groups in the verse).  Ken Graham already covered the OT view of polygamy, so I will not repeat it here.  Ken also covered how the criteria of elder's marriage shed light on monogamy, so I will not repeat it here either.

The NT support for rejection of polygamy is based on what Jesus said in Matt 19:3-9 as the original intent of marriage before the Fall.  I think in general Christianity agrees that marriage is to be a symbol of God's covenant between Jesus the bridegroom and The Church the bride, prefigured in the covenant between God and Israel.  Read the whole book of Hosea which revealed the exclusive love that God demanded from Israel as a proper response to God's exclusive love for Israel, His chosen people.  Hosea is a very touching book of how God STILL wanted Israel to come back to Him, even after Israel did "polygamy" and "adultery" by simultaneously worshiping other gods.  The theme of monogamous relationship continued on to the NT period where there is only one Church and one Jesus.  So it makes sense that the covenantal relation between husband and wife should be modeled after the relation between Jesus and The Church (see Eph 5:21-33)

This booklet on The Bible's Teaching on Marriage and Family (pdf here) is an excellent, easy-to-read, theological exposition on why Christianity espouses monogamy as well as the purpose of Christian marriage and how Satan wants to destroy it.  The booklet contrasts between the secular notion of marriage as contract versus Christian notion of marriage as covenant.

On the other hand, NT teaching on widow remarriage is more on a prudential basis, far less dogmatic than the teaching on monogamy.  The main relevant passages are embedded on Paul's recommendation to the churches he founded in Corinth (1 Cor 7 esp. vv 39-40) and Ephesus (1 Tim 5:11-16) so that the members of the church can grow toward holiness (you can get this sense from the circumstances which precipitated the 2 letters).  Thus, Paul's advice on widow remarriage is to prevent distractions, temptations, and sins from polluting the local church body (example: like bad yeast in 1 Cor 5).  Notice the overall context of the passages: the importance for a man and a woman to channel their desires into wholesome patterns instead of into deviant ways that the enemy (Satan) can use.  This is because one theme of the overall teaching of Paul is the importance of holiness so Christians can enter the Kingdom of God (i.e. receive their promised inheritance) on the Day of Judgment (see 1 Cor 6:9-11, Gal 5:19-21, Eph 5:1-5).  

See also a gotquestions.org article on What does the Bible say about remarriage after the death of your spouse? commenting on the same NT passages plus some OT background, saying that widow remarriage is "absolutely allowed by God".

The care of older widows (1 Tim 5:16) who cannot provide for themselves was a carryover from the OT ethic for helping the weak, which became the basis of why God condemn the powerful Jerusalem elite who instead of using their power to protect the weak they use it to oppress "the widows and the orphans".  So Paul's teaching about widow remarriage should also be read in the immediate context of how the local church should choose which widows to support.

In modern North America (USA, Canada) and Western Europe, the situation has much changed since the NT period since: 


women now have equal rights and opportunities to pursue their career
there is not much social stigma to single career women
there is nursing facility and social security payments for older widows
household arrangement is more nuclear than extended, so less "temptation" from busy-body younger widows living in the same household


Therefore, the churches have a lot less responsibility to care for widows than in Paul's period so this impacts the prudential judgment about whether widows should or should not remarry.  But there is still a lot that we can do today.  This scholarly article titled The Order of Widows: What the Early Church Can Teach Us about Older Women and Health Care describes the practice of the early church in light of Paul's teaching in 1 Timothy as well as the OT practices mentioned above as a background to what we can do today to help modern elderly women.

The NT is rather silent on the reunion of the marriage bond in heaven.
Two frequently cited NT passages on marriage in heaven are Luke 20:34-36 and Mark 12:18-27 with sample interpretations in Why is there no marriage in heaven? and Marriage in Heaven? Will We Know and Love Our Spouses in Heaven? respectively.