As littleadv says, how the income tax exemption from the treaty works depends on exactly what the treaty says.
But for whether you are a resident alien or nonresident alien for U.S. federal tax purposes, that is determined completely by U.S. domestic law, and is a separate issue from treaty tax exemptions. Who is a resident is discussed in Publication 519.
You are a resident alien if you satisfy either the Green Card Test or the Substantial Presence Test. For the Substantial Presence Test, you are a resident for 2014 if (the number of days you are present in the U.S. during 2014) + (1/3 of the number of days you are present in the U.S. during 2013) + (1/6 of the number of days you are present in the U.S. during 2012) >= 183 days. Basically, if you are in the U.S. for half of the year, you meet the test, and even if you aren't, days from previous years can be counted in a certain proportion.
However, days are not counted in the Substantial Presence Test for people in certain statuses, the "exempt individuals". You didn't say what kind of J1 you have; if you were a J1 student, then you would be an exempt individual as a "student"; for any other kind of J1, you would be an exempt individual as a "teacher or trainee". However, whether you are an exempt individual also depends on whether you were an exempt individual in past years. A "student" is not be an exempt individual if they have been an exempt individual for any part of 5 previous calendar years. A "teacher or trainee" is not be an exempt individual if they have been an exempt individual for any part of 2 calendar years in the previous 6 calendar years.
Basically, if this is the first time you're in the U.S., this means that, if you are a student, you will be a nonresident alien your first 5 calendar years, and a resident alien thereafter; if you are a J1 and not a student, you will be a nonresident alien your first 2 calendar years, and a resident alien thereafter. So, for you:
If you are a J1 student, you are an exempt individual for your J1 time in 2014. None of your days are counted towards the Substantial Presence Test, so you don't pass it. You would be a nonresident alien for all of 2014.
If you are a J1 and not a student, you would not be an exempt individual for any part of 2014, since you have already been an exempt individual for some part of 2 of the last 6 years (2012 and 2013), and thus all of your days in the U.S. in 2014 are counted in the Substantial Presence Test, which you would pass for 2014. So you would be a resident alien for all of 2014.
In neither case would you be dual-status.
In many tax treaties, the tax exemptions do not apply to resident aliens, due to a "saving clause". If your tax treaty is like this, and if you are a resident, then you are not exempt from the tax. However, some tax treaties do not do this; e.g. the U.S.-China tax treaty's saving clause does not apply to the sections for students and researchers, so they can continue to use tax treaty benefits after becoming a resident. (The U.S.-China tax treaty has 36 months of income tax exemption for researchers, which usually falls over 4 calendar years, much longer than the 2 calendar years that they are nonresident.)