I cannot remember where, but somewhere in gemara it mentions that when confronted with doing a positive mitzva or keeping a negative mitzva, one should do a positive mitzva. How far does this go though? What are some negative mitzvot that one cannot do (outside of those three cardinal sins) that one may break in order to do a positive mitzva?
The rule is עשה דוחה לא תעשה - a positive commandment pushes away a negative commandment. Basically this is in fact quite limited, but in theory if something is simply forbidden by a negative commandment but a Mitzvah comes along that requires violating it in order to fulfill it, then we go ahead and do the Mitzvah if there is no way to avoid the negative commandment. (If it could have been avoided previously but can't be right now when it is time to do it is a subject of dispute).
Not all negative commandments are included in this dispensation however, and sometimes Chazal forbid it (as the case of a Yibbum where there is a negative commandment involved in the relationship).
Examples of excluded negative commandments include theft, things which are forbidden by a positive as well as a negative commandment, as well as possibly including things which have a death penalty associated with them, where there are two negative commandments involved simultaneously, and other such exceptions.