Sanhedrin 4:2 teaches that in a monetary case or one involving (im)purity, judges give their opinions starting with the most senior, while in a capital case they begin with the most junior. In the g'mara R' Aha b. Papa says one should not speak 'al rab, against the chief of the judges. Apparently, contradicting a judge senior to you who has already given his opinion is a problem.
I understand why we want to be extra-careful in capital cases, but if we are concerned about a senior judge swaying others in a capital case, shouldn't we also be concerned in a monetary case? The Torah is very clear that judges must judge fairly -- always, not just for certain cases. So why do they not always start with the juniormost judge and work up, instead of having two different orders depending on the type of case? It doesn't seem like a hardship to do it this way; everybody is going to speak either way and it's just a matter of ordering.