I just logged out of MN (for the first time in months, I had to search the page for the logout button!) to conduct some research. If you close the page after logging in by touching where it tells you to, you do go back to the page telling you to log in, but it then refreshes automatically & gives you the search page. The only way I can make it not work is by using the multiple page icon & closing the page with the red cross.
Thing is, I do seem to remember having trouble when I first got my iPhone, but I can't remember what I did - I probably just logged in on the PC! Since then I haven't had any problems, but then I rarely log out anyway!
MNHQ did say something about a mobile/iPhone/Blackberry version, don't know how high that it on Tech's List....
Actually I do have one more suggestion - go to Settings on your iPhone & scroll down to Safari. Look at your Accept Cookies setting - mine is From Visited - if yours is set to Never, it may be that it's blocking a cookie that your login needs.