All quotes in this post are taken from "A Chance to Start Over: Church Disciplinary Councils and the Restoration of Blessings" by Elder M. Russell Ballard (an LDS apostle) in the September 1990 Ensign, an official LDS publication.
There are two levels of discipline in the Church: informal discipline (at a Bishop's discretion) and formal discipline (which is recorded). Formal discipline requires a disciplinary council.
When a bishop learns of a transgression, usually through the confession of the member involved, he first counsels with the member. When the sin is not grievous, the bishop may decide, through inspiration, that no disciplinary action is needed. He may continue to give counsel and caution, helping the member resist temptation and avoid further transgression.
Another option the bishop has is to place the member on informal probation, temporarily restricting his privileges as a Church member—such as the right to partake of the sacrament, hold a Church position, or enter the temple. The bishop may ask the member to surrender his temple recommend temporarily. In addition, he may require the member to make specific positive changes in attitude or behavior. No official record is made or kept of informal probation. The bishop maintains close contact with the member and may terminate the probation period when he is prompted to do so.
Decisions of [a disciplinary council] are to be made with inspiration. A council can reach one of four decisions: (1) no action, (2) formal probation, (3) disfellowshipment, or (4) excommunication.
Non-payment of tithing only merits informal discipline:
Disciplinary councils are not held for such things as failure to pay tithing, to obey the Word of Wisdom, to attend church, or to receive home teachers.
When are such councils held?
The First Presidency has instructed that disciplinary councils must be held in cases of murder, incest, or apostasy. A disciplinary council must also be held when a prominent Church leader commits a serious transgression, when the transgressor is a predator who may be a threat to other persons, when the person shows a pattern of repeated serious transgressions, when a serious transgression is widely known, and when the transgressor is guilty of serious deceptive practices and false representations or other terms of fraud or dishonesty in business transactions.
Disciplinary councils may also be convened to consider a member’s standing in the Church following serious transgression such as abortion, transsexual operation, attempted murder, rape, forcible sexual abuse, intentionally inflicting serious physical injuries on others, adultery, fornication, homosexual relations, child abuse (sexual or physical), spouse abuse, deliberate abandonment of family responsibilities, robbery, burglary, embezzlement, theft, sale of illegal drugs, fraud, perjury, or false swearing.
Thus, according to official LDS policy, members can only receive informal discipline (such as loss of a recommend) for not paying tithing.