Blender version: 2.79
Side notes: blender also does not like Inline nodes for x3d; will try with the newest Blender version again; blender could also have some issues with exporting fbx humanoid models, investigating
Per this page (relatively maintained), blender does not support exporting key-frames for x3d.
Blender uses add-on scripts for exporting. The default importing/exporting add-on for x3d is called "Web3D X3D/VRML2 format". The documentation page shows that, at least till Blender 2.76, a lot of features for this add-on is still unfinished. The page does not mention animations specificically though.
Tools: Blender v 2.79, a moving and rotating cube scene in blender: avalible here.
Export output. No interpolator/TimeSensor node can be found. Here is the result after importing back into Blender.
Try add a simple rotation to the previous file. Can be varified through Instant Player that it works fine.
However, when imported back into Blender, the animation disappears again.
It can be expected that, complex humanoid models (or this version) have the same issue.