|Hi Trent -
Interesting question and I have to answer with "it depends". First off, a reaction to the urushiol oil contained in the poison ivy plant is not an infection. It is an overblown immune system response called allergic contact dermatitis. There certainly is the possibility of a secondary infection if a rash is improperly cared for. The terms are important for as worded, your question would imply an infection requiring professional medical treatment (possibly with antibiotics). I'm assuming you mean the "garden variety" of outbreak without a secondary infection.
Now here is the "it depends" part based on my assumptions above. Your sensitivity to urushiol oil will play a large part in the duration of the reaction. The rash varies from person to person and from time to time. The rash I had last year took 6 days to heal but the monster rash I had a couple years ago took 3 weeks to heal. Same person (and true incidences here) but different durations.
Here is another factor that adds to the "it depends" category - when and what treatment is used. If you are placed on steroids BEFORE the outbreak of the rash, you can (possibly) short-circuit the response and prevent a rash. This is an unlikely scenario as most folks don't know until the rash arrives that they've been exposed. Steroids can possibly also prevent the rash from getting worse and thus keeping the reaction to a minimum - there are issues and side-effects so while this is an effective treatment for many, it doesn't work for everybody. Another factor to consider.
Scratching that itch? Can add time to the duration of the rash. Stressing over the rash? Can possibly add time to the duration of the rash. Using an inappropriate treatment (such as harsh chemicals on open wounds) - can add time to the duration. There are many factors here which affect the duration of the reaction to urushiol oil.
Without knowing what treatment you are interested in, I couldn't even give an uneducated guess. So here is the norm - treated 7 to 14 days, and untreated 1 to 2 weeks.