1. They're playing the violin.
2. They're playing the violins.
Which of the above sentences is correct?
He/she is playing the violin.
Even so, if all three people were holding and playing a violin, you would still say "They're playing the violin", or "They're all playing the violin". There wouldn't be any confusion because three people wouldn't all be playing the same violin (possible but unlikely!)
However, if you remove the definite article "the", then you would be able to say:
They're playing violins.