What is convolution of two sine waves (tones)?
Convolution in the time domain is equivalent to multiplication in the frequency domain.  

If you window two sinusoids in the time domain to get finite length waveforms, and the two sinusoids are exactly integer periodic in the window width, then the DFT will be impulses.  If the frequencies are different, the impulses in the two DFT will be disjoint, and the multiplication of the two spectrum will result in zero output.  

If one or both of the two sinusoids are not exactly integer periodic in your window width, then the FT will result in a Sinc function.  Then the convolution will be equivalent to multiplying a Sinc function against an impulse or another Sinc function, which will produce a non-zero result.