"Breathes" is not a very meaningful term in that wood doesn't have lungs. But briar, like all wood, is constantly exchanging moisture with the ambient air around it. In a dry climate, briar is more dry inside than in a wet climate. This is why cabinet doors are made with panels, it's why things in your house shift and stick at certain times of the year. It's why table tops are built with sliding hardware to acommodate the wood's seasonal movement. You can stabilize wood by vacuum pressing cyanoacrylate into it. But that's not done on tables, nor to pipes.
OIl cured briar would likely exchange less moisture, or rather, exchange it more slowly, than regular briar, but briar absorbs and releases moisture very slowly in any case. This is why a pipe is not 1/2 pound heavier after smoking.
I've posted this picture before: You can see the moisture "boiling" out of a block as you drill it. Happens with every pipe, every block. That moisture will be re-absorbed (from ambient humidity in the air) over time.
Just before drilling:
You can see a ring of "dark" just around the chamber as it is being cut. That's water coming out.
This same thing happens every time you smoke. You heat the pipe up, and ambient moisture in the wood tries to escape. If you have a thick vapor-proof finish, it might blister, and we see that on certain pipes from time to time. If you have a thin finish that is not very vapor-proof, like shellac or a little oil (tung oil is more vapor proof), you'll see nothing.
Does it affect how a pipe smokes? Nah. No more than staining it black vs brown.
Most smokers have no idea how most of the pipes they own were finished. If the finish doesn't fail, they assume it's one kind of finish or another, some "approved" finish. They'd be shocked to learn it's lacquer, shellac, tung oil, acrylic, or even polyurethane.