Hi!
In case anyone has similar problem.
I had similar issue with mixamo rig. Everytime I saved the project and opened it up the mesh was back to T-pose and didn't move with the rig.
I tried to upload different formats to mixamo (FBX and OBJ) to see if that makes any difference but it didn't. Among dozens of test rigs that I made I noticed that one was actually working even after reopening the project file. After further investigation it turned out I saved the project file when the time marker was in the middle of the timeline and it opened with the marker in that same place.
When you import the animation from mixamo and rig it in C4D try saving it with time marker not at frame 0. Let me know if that worked for you.
Thanks!