Referring to the deep end of the swimming pool. Most people learning to swim will start where they can touch the bottom and keep their head above water. Others will jump in at the deep end and sink or swim! If you want to learn to be a food server, starting on the busiest night of the week is jumping in at the deep end. If you want to learn to raise rabbits and start off with a hundred of them, that is jumping in at the deep end. One would hope you had received instruction before doing any of these things. So, while it does imply some courage and risk taking, it does not have to mean you are irrational.

"Taking the plunge" has the implication of doing something that you know carries a risk. Doing something that is risky is always a little irrational because you depend on luck for everything to turn out well. So the amount of "acting irrationally" the idiom carries depends on how much risk is involved in the situation it is used to describe.

No, it does not mean to act irrationally. It means to get engaged. (Some may think that is irrational, though.)