In an orchestra scenario, under what circumstances, when the piano and violins are tuned to 440Hz (as shown on the tuner) together, will the harmonic series between the two differs and make the sound produced less harmonious?
In acoustic instruments, the fundamental is always stronger than the harmonics so this is an extremely implausible scenario during live performances.
However, subharmonics can occur if the intonation is off, thus resulting in a dissonance that deviates from the intended tone. ()
As has already been said in the comments on the question, any difference between the harmonic series of tones on the piano and on violins has nothing to do with the agreed upon pitch of A.