The quickest and easiest way to fix this in this case is as follows:
- Select all of the shapes that make up your logo
- Duplicate them as a unit (copy and paste in front for instance)
- Combine all of the components into one shape using Pathfinder, Unite
- Colour this combined shape the main flat red of your logo
- Send it to the back
Screenshots below show the results with and without the red shape at the back:
For future reference, it is better when building logos like this to start with a shape that makes up the overall form of the logo and then build other objects on top of that shape, rather than having lots of slices that ‘kiss’ up against each other.