Judaism has always regarded the misuse of public funds to be a most grievous sin. Whereas Jewish law stipulates that a worker be given opportunity to correct his inefficiencies and be given ample warning before dismissal, one is mandated to remove immediately and without warning those who violate the public trust. Additionally, due to the difficulty of properly reimbursing those who were wronged it is doubtful whether teshuva (repentance) is even possible. It is for this reason that when electing public officials, be they in our shuls, schools or on government benches, we should follow the advice of Yitro and seek "men (or women) of accomplishment, G-d fearing people, men of truth, people who despise money" (Exodus 19:21). It is due to the extreme importance and difficulty of properly serving the public properly that every Shabbat we offer a blessing to "those who involve themselves in the needs of the community faithfully".
The nature of public work is such that hard painful decisions are often necessary, decisions that will inevitably hurt some people. While the faithful public servant is exonerated as it is a necessary by product of the mitzvah of serving the public; absent the pure motivation of 'acting for the sake of heaven' one must bear full responsibility for the pain caused. This is a burden that is truly a heavy one and public life should be avoided by those who cannot meet this standard.
The Shulchan Aruch rules that one should not initially appoint those who have a shady reputation even if they relate to events of years past. The stakes are too high to hope that the rumours are not true and that the person has truly changed. While this ruling is codified in the laws of appointing religious functionaries it appears that the same rules would apply to all public servants. However once a person is already serving in a public capacity they cannot be removed on accusations alone. Hard evidence i.e. testimony of two honest witnesses is necessary. The Chafetz Chaim - who spent his life stressing the importance of not speaking or even believing lashon hara - adds that a credible persistent rumour of wrongdoing, provided it comes from those who have 'no axe to grind' may also be grounds for dismissal. However for a sitting public servant who has done teshuva for past wrongs and is currently performing properly dismissal would not be allowed.