I think the problem stems from the .fbx importer not by default treating the eye's image texture as having a transparency channel (an Alpha channel, in addition to the RGB color channels), although I am not sure why this would happen on some but not other models downloaded from the same site and created by the same modeler. Still, this seems to be the cause in the particular case of the Dragonite model you downloaded.
By default, this situation results in an import that renders like (note that this is in Eevee; the problem exhibits distinct visual problems in Cycles, but presumably the fix would be similar there, accounting for the somewhat different handling of Transparency in Cycles):
If we open a Shader Editor window, and select the eye material (an image texture that is mostly transparent, except for a small section in which the eye is painted; note that this is likely to be the last of four Materials on the Dragonite model, for you), we see something like:
There are two issues here:
- The material's "blend mode" is set to Opaque, which does not use the Alpha Channel, and so ignores transparency.
- The Color rather than Alpha output from the Image Texture has been connected on fbx-import to the Principled BSDF Shader Node's Alpha input channel, so Black-White is being interpreted as Alpha. If we fix these problems (see squared areas in this image), the problem appears to be fixed: