I agree with @janbb and @ibstubro. It actually means stupid but to a lesser degree. It usually would mean foolish and, or ignorant, depending on the context.
Example. If a girl caught her boyfriend cheating, she may say I feel dumb. Not really meaning that she thinks she is stupid but foolish for believing in him.
Example. An adult in an amusement park ride that is meant for children. May say I feel dumb and they may mean silly if no one they knows witnesses it or foolish if someone witnesses it and mocks them for it.
Example, A kid saying I feel dumb after failing a rather simple test. Means they feel stupid.
Example, Someone quoting something they believed to be fact and find out later that the information has changed since they learned it , may say I feel dumb and what them mean is that they feel ignorant.