After you have washed and patted dry your arm pits try using some talc or baby powder under there until your burning/itch is gone. You may be having an allergic reaction to the deodarant you are using and will need to try a few different ones until you find one that doesn't cause this. If the same thing happens when you use the powder too you will need to find something for very sensitive skin and hypoallergenic also nonscented.
I would suggest avoiding the talc or baby powder since they contain unnatural substances. Best thing that I can suggest is to go see a doctor, you may have a bacterial or fungal infection.