I believe Muirsheen is a name. I suppose it could come from Máirtín,a s has been suggested, but I always thought it was "Muirisín", little Maurice. But who was he - landlord, boss, moneylender (or have I got it wrong and Mursheen was a relative or friend)?
I had a notion (without firm basis) that like 'Nora' and 'Many Young Men of Twenty Said Goodbye', this song was part of a stage play - and then it would be clear who Mursheen was. But according to Seán McMahon (see M. Ryan's message) the song is from a 19th c. broadsheet.