He becomes responsible the minute he steps into office. But the more important question is
what he is responsible for. To me, he and every other president or chairman is responsible for the extent to which his policies made a situation worse under circumstances when it should have been made better. Or for the extent to which his policies made a situation better which ought to have been made much better.
You pointed out a while ago that Nixon should not be held responsible for our presence in Vietnam, and you were right. It was an unholy mess when he took over, and he was not going to magically change that.
Clinton took over with a deficit and left with a surplus, but his policies also contributed to the lending offenses that were a big part of the economic collapse, so Bush should not be held solely responsible for the financial disaster he left us with either. To be sure, he made it worse, and the continual right wing chorus of keeping government out of business led to serious lack of oversight of the banking industry (my under educated and uninformed opinion). But we liberals deserve criticism for laying it all on Bush.
So Obama definitely deserves to be held accountable for that part of the economic problem that reasonably should have been resolved or improved by now. A while ago, I posted a question asking posters to identify who would have guided us to a better situation with the economy and with what policies. No answer. Our coping mechanism for dealing with the size and complexity of fiscal policy issues is to just dig in and wallow in the most simplistic and useless generalizations. I do it.
Something like Iraq? Bush's baby, no doubt, though most politicians were just too chicken to let wisdom prevail over overheated and much abused appeal to patriotism. Whoever inherited the mess Bush left in Iraq was going to have to deal with the kinds of problems Cheney warned about. If Obama fails to apply policies to improve the situation or gets us in a worse situation when another policy could foreseeably have avoided that, it's on him.
Short answer, though? Never.