Upon addition of dimethylamine on dimethylaniline, there would be an intermolecular hydrogen bond; where hydrogen atom is located between 2 highly electronegative atoms (i.e. Nitrogen), and a bridge betwen molecules would be formed.
This, in turn, will be followed by an increase of boiling point of the mixture.
Another result of this addition, basicity of the mixture will increase.
Above all of these, I think there is no chemical reaction between them; they are two amines!!!
:) :) :)