Do climbing shoes need to be broken in?

All climbing shoes have a break in phase. During this phase, the climbing shoes will stretch and get slightly softer in some areas. The amount at which they will stretch and get softer depends on the size of the shoe, the material, and how you climb in the shoes.

If the shoes are synthetic, the break in period may be shorter than that of shoes with leather since synthetic shoes normally do not stretch that much, they usually stretch around half a size to one size. This means that if you buy a size 9 synthetic climbing shoe, it will most likely still be the same size and fit as you bought it, even after many months of climbing in it.
With synthetic shoes, though, It's important to keep in mind that they will get softer after consistent use, but, since they won't change in their fit too much, and they will be almost as tight as you bought them even after 6 months, it's important to really get the correct size and to choose shoes that fit your feet properly. Otherwise, the break in period can take much longer, and may not happen at all if the shoes just don't fit your feet.

If the shoes are leather, the break in phase is a little bit different, since leather shoes can stretch up to a few sizes larger than when you bought them.

This is from personal experience of having 4 pairs of synthetic climbing shoes, all of which are as tight as they were when I bought them and have not stretched (from what I can feel at least) even half a size, and this is after between 2-3 years of consistently climbing in them 4 times a week. They have become softer to some extent, which reduced the feeling of tightness in some areas of the foot. The break in period for me was relatively non-existent, since they didn't really stretch, and I bought them with a pretty good fit out of the box. They started getting softer, I would say, after about a month of consistent climbing,