In most New Keynesian models, can monetary shock affect technology shock also? One simple New Keynesian model would be http://crei.cat/people/gali/pdf_files/monograph/slides-ch3.pdf but I do not restrict my question to this model only.

Let us for now ignore the problem of linearizing around zero-growth-rate, and assume that there is no problem in modelling constant natural growth rate.

Let us suppose that monetary shock occurred, due to stochastic or forecast errors or intentional deviation. Can this shock result in technology shock also? (ex. change in $A_t$ in production function?)

I am sure that there exists a model that does this, but in most models, and especially Gali's model as linked above, is this true?
No it usually does not.

There is an intuitive reason why: We interpret the technology shock as a residual, the unexplainable part in TFP changes. If we knew that there was a specific causal relationship between monetary policy and future TFP, that part of future TFP would now was explained. 

It's true that the MP shock is excluded in standard Solow Regressions, but if you believe there is a causal effect, you need to regress it out of it. Since after all, we understand $A_t$ to be independent.

Note that still, implicitly, through its effects on capital accumulation (which does not exist in standard NK models), monetary policy should impact future TFP. Then, you could make $A_t$ dependent and argue it being a shortcut in the absence of capital.