If none of the following descriptions matches your problem, please feel free to report an issue.
Animation (import and export)
Although simple skeletal or keyframed animations should work fine, more complex animations could cause unexpected behavior.
There is no "quick fix" for those kinds of behaviors, which are actively being worked on our side.
Here is a list of known issues on import, as well as some possible fixes.
Please note that the materials are being converted from Sketchfab to Cycles in Blender 2.79, and Eevee in Blender 2.80. If a material looks wrong, using the Node editor could therefore help you fixing possible issues.
Empty scene after import
Scale can vary a lot between different models, and models' origins are not always intuitively centered.
The imported models will always be selected after import, and you can try to scale them in order to make them visible (most often, the model will need to be scaled down).
If it's not enough, try to select a mesh in the Outliner view and use numpad '.' (View to selected operator) to center the view on it.
Some models are using refraction on Sketchfab (for glass, ice, water...), which is not supported by glTF and ends up being converted to regular transparency.
In Blender Node Editor, refraction can be achieved by tweaking the IOR and Transmission inputs of the Principled BSDF node, or by mixing a Refraction BSDF shader with the original material.
Weird seams or normals
Tangent space import is not working yet so you might experience rendering issue on some models with normal maps.
Single color model (wrong backface culling)
Backface culling is not well supported on import yet.
It is often used on Sketchfab to create models with outlines (as on this model for instance) by duplicating the object, scaling it up, flipping its normals and making the material single sided.
You can reproduce this behaviour in Blender:
For the 3D view (not rendered), check the Backface culling checkbox in the Properties Panel (shortcut N), under the Shading dropdown.
For the rendered view (in Cycles), follow the instructions on this StackOverflow answer.
Blender 2.80: In the node editor Properties Panel, under the Options tab and Settings dropdown, make sure to have the Backface Culling option toggled on.
Unexpected colors (vertex colors)
If your model displays strange color artifacts which don't seem to be caused by textures, you can try checking the model's vertex colors information (Properties area -> Object data tab -> Vertex Colors layer), and delete the data if present.
Vertex colors are indeed always exported in glTF files (to allow edition), and always loaded in Blender. It is possible that this data is corrupted or useless - but disabled on Sketchfab - explaining why the online render looked fine.
Some transparency settings might not be processed correctly, and just using a Transparent BSDF shader or linking a texture to the Alpha input of a Principled BSDF node might not be sufficient: try to set the opacity settings in the Properties Panel of the Node editor, under the Options tab by setting the Blend Mode to Alpha Clip or Alpha Blend.
In some very specific cases, the processing of your model can crash due to "heavy" textures.
If your model does not process correctly in Sketchfab and that you are using multiple high resolution textures (for instance materials with 16k textures or multiple 8k textures), you can either try to reduce the original images size or upload your model without texture and add them later in Sketchfab's 3D settings.
As of now, textures colorspace set in Blender are not automatically converted to Sketchfab, and although normal maps, roughness, metalness and occlusion textures should be processed correctly, setting a diffuse texture's colorspace in Blender as "Non-Color Data" or a metalness map as "Color" (sRGB in 2.80) will be ignored.