Keima is using the liver as a metaphor. In that scene, you can see a diseased liver on the screen. That shows that the liver has to be operated on. That operation will lead to a change so that she can live. The liver, being an internal organ, is a metaphor for her inside voice. Keima is using it to illustrate how though Shiori is quiet on the outside there is a lot more going on the inside and significant changes could be taking place with out people realizing it. Keima needs to perform an "operation" so that Shiori will be able to express herself and "live".
Why they chose to use the liver is uncertain and never explained. Please look at nhahtdh's comment as it does provide some insight into this.