I haven't heard this particular song, but generally the expression means that everyone in the audience is crying. The expression is used when a performance is so sad that it brings an entire audience to tears. ("House" is a theatrical term for the audience seating area).

Hi Luis Not a dry eye in the house, is an expression used when describing a situation where people are overcome with emotion and have tears in their eyes or are crying. It can be for both a happy or sad situation. I hope this helps