Every so often, at the conclusion of our bed bug dog inspections, a client asks if bed bugs can live on and be transported to and from their pet, cat etc.
Although I am not an expert on this issue, I have taken an entomology course in my younger years and so I can understand the why and why-not of certain evolutionary behaviors of insects.
Unlike a flea which is relatively flat in a “vertical” plane and thus can easily navigate around hair and fur, a bed bug is flat in a horizontal plane and has a difficult time maneuvering around hair and fur. Therefore bed bugs prefer “naked”, hairless skin over hairy and furry ones.
This is not to say that it could happen as for instance, in a high infestation environment where there is competition for blood source.  Or when human blood, their meal of choice, is not available for an extended period of time and a cat or a dog would be an option for survival.
So, my answer is:  Not likely so, although it could happen.
If you are interested, you can read more about this subject in this article: Bed bugs and cats.