I did a fresh install (~/.minecraft/ did not exist yet) with the new launcher and I made sure the "remember me" switch was on when I logged in. Everything installed normally, and the game worked fine. But when I closed the game and tried to open it again, it asked me to login again. After that point, it correctly kept me logged in. I tested this several times (by removing the ~/.minecraft/ directory and then reinstalling) and the problem happened every time. I also tested this on Windows, and the problem does not appear to happen there.