I think there are songs that sound like they are attempting to copy The Beatles or are Beatlesque. I have never heard a song that I thought, wow, that sounds like it could be The Beatles. In my opinion The Beatles have a sound that has never been successfully copied. Exactly where their "sound" came from I don't know. I think it was many factors. George Martin, Geoff Emerick (and the others... Normal Smith, etc.), the songwriting of Lennon/McCartney, Abbey Road Studios, the instruments that they used, and more. I think it was/is a complex formula that can not be imitated or repeated. I love the new (1995/6 Anthology) songs, but even they do not have The Beatles sound completely. There is a little something missing, and you have all four Beatles playing on them. Which tells me that it had to do with studio, producer, engineer as well.
Many factors. I know the title of this thread was which ones are close.
The Rutles did a good job, "Get Up and Go" sounds a lot like "Get Back" and The Knack's "My Sharona". In 1977 when "My Sharona" came out there was speculation that it was The Beatles. I don't think it sounds like The Beatles, but I will say it is "Beatlesque".