Disadvantages of a republic government include that it requires too many people to make a decision and that it was slow to render a decision. It also occasionally resulted in dictatorship, and wealth still dictated power within republic society.
The basis of a republic government is that it is ruled by the people. While serving as the basis for modern democracy, the requirement of multiple officials in order to reach a single decision could be problematic. This was especially true if there is a decision requiring very swift judgment. The Roman republic response to this was to elect a temporary dictator to wield total power for a period of six months. The problem with such an arrangement is that some did not willingly relinquish total power after being given it. The most famous example of this is Julius Caesar, who held the six-month position of dictator for five years before being killed.
In a republic, as in most governments, one problem is that the wealthy can still purchase the influence of elected leaders by employing their wealth. For the Roman republic, this goes back to the strict societal divide between the aristocratic patricians and the lowly plebeians. Even as the institution of Roman republic government allowed plebeians equality and a chance to rule via the consul, wealthy patricians were still able to effectively control politics through their wealth.