Personally, I drop "the." That being said, both are correct. It depends entirely on the person. I do play guitar and piano, though.

You use "the" to imply "out of all musical instruments". If you drop "the", we can still understand it, but you will sound unnatural.

Yes, I agree with Akahitoha. You can say it both ways, but I think it definitely sounds more natural to include the "the": I play the piano, I play the violin, I play guitar.

I'd suggest saying "the" with it, but some people do drop it. Generally it depends on how you're saying it. "What instrument do you play?" "I play the piano." "What instrument do you play?" "I play piano". It's basically the same in both cases, the "the" was just dropped because the topic was understood. Technically, to be grammatically accurate, you should include the "the", but some native speakers choose to leave it out.