It is worth noting here that sacrifice, by definition, must be giving up something we value. The ten percent of annual increase to be paid in tithing each year carries very different values for a rich man and a poor man. Though the dollar amount may be higher for a rich man's contribution, that rich man may value money less than a poor man because he has more of it. So, while the poor man may be missing meals to pay tithing, the rich man, who recovers much of his contribution in his tax return anyway, may hardly notice the transfer of money to the Church.
On the other hand, time for the wealthy man may be much more valuable than for the poor man. Certainly the rich man has more to lose for every hour of work missed than the poor man, for example.
The Church helps each of us to sacrifice things that we value in order to make our worship more true and our blessings more plentiful. She may ask a poor man to donate to fast offering, humanitarian aid and missionary programs in addition to his tithing to reach this objective. Similarly, she may ask the rich man to serve in a time-intensive calling such as a bishopric or as a scoutmaster. One thing the Lord has asked all members to sacrifice is a broken heart and a contrite spirit (D&C 59:8).
The Church has pushed its members to sacrifice worldly things for the greater blessings of full activity in the Church. For the earliest Saints, this meant conversion despite the persecution of mobs. In the next generation came leaving civilization for a trek across the prairie and starting new settlements in the West. Soon came the end of polygamy (not particularly worldly, but a difficulty at the time that even led to the excommunication of a member of the First Presidency), then the emphasis on tithing, then enforcement of the Word of Wisdom. Currently, the Church's emphasis on chastity issues provides a significant opportunity for the members of the Church to sacrifice the things of the world around them for the greater blessings of God. Additional sub-emphases, including missionary service for men, temple attendance, daily scripture study and a long list of others have been brought up from time to time as well. Sacrificing in these ways helps each of us to become more unified Saints and more worthy of a life with God.
The blessings of sacrifice to the individual and to the Church are tremendous. Secular studies on religion show that high demand of church members increases attendance, growth, donations, health and marital stability while reducing free-riding and juvenile delinquency (all sources cited in Iannaccone's Introduction to the Economics of Religion, 1998, pp. 8-9). If that short-but-action-packed list isn't enough, countless, less measurable blessings remain including humility, sociality and perspective.
In short, sacrifice is a gospel practice and covenant that assists in the perfecting of the Saints. The Church has always encouraged sacrifice from its members to motivate true worship, full conversion and individual health and happiness. Living the law of sacrifice brings tremendous blessings upon the Church and the individual.