What if our axioms are false? What happens then?
If an axiom in a deductive system is contingently false (i.e. could be true, but isn't), you can come to some false conclucions. For example, if you axiomatically assume all mammals are viviparous, you can come to the conclusion that momotremes are viviparous, which they are not.
If an axiom in a deductive system is logically false (i.e. cannot be true for logical reasons), you can come to any conclusion you want. This effect has been known in the latin ancient world as "ex falso (sequitur) quodlibet (EFQ)". The reason is the way the deductive argument form of modus ponens works. From the axiom "A and not-A" you can virtually conclude anything!
In an axiomatic system that deals with abstract objects -- think of mathematical systems -- it might not from the beginning be obvious whether the single axioms are true or false. It may then, in the process of deducing, by contradiction turn out that they are inconsistent, i.e. that not all of them can be true together. Then you are back to one of the two cases above.
"Wrong" is not the correct term. We'd simply rather axioms not be "Inconsistent".
This happened to be the case with the axioms of Cantorian Set Theory. It was found to derive several contradictions, or paradoxes, such as the infamous Russell's Paradox.
As a result, the axioms of Cantorian set theory were formalised and refined in the hopes of eliminating such paradoxes; resulting in the axioms of Zermelo-Fraenkel Set Theory.
So that is what happens when the axioms of a theory are inconsistent.