"If you keep singing Phish's songs" this means the song belongs to Phish and no one else. "If you keep singing Phish songs" this means that the song is from Phish. Here Phish is used as an adjective to describe the type of a song. In this case, from the context, a song created by Phish. Big difference in meaning. Take another example, The blue sky. Here "blue" is the adjective describing the color of the sky. The blue's sky. Here the sky belongs to blue. However this makes no sense if blue means the color. However, if it is meant to be a kind of music, then it makes sense.

I think "If you keep singing Phish" and "If you keep singing Phish songs" sound the most natural. I don't have a good explanation why, but it sounds unnatural to use the possessive form when talking about a song the band wrote. Example: "This is a Phish song." We will also use this same grammar if we are speaking about a concert or show. For example: "Did you see the Phish show last weekend?" "I'm going to a Phish concert tonight."