It depends on how much tartar was there in the first place, whether the dentist had to perform deep scaling and whether there was a lot of swelling in the gums before treatment..
You may feel that the teeth have become mildly loose, but remember that's just a "feeling".. This happens especially whether there has been huge amounts of tartar and consequent gum recession..
You may also experience increased sensitivity due to exposed root surfaces which were previously covered with tartar..
Next Steps
After cleaning, avoid eating anything too hot or too cold for a couple of days.. Avoid brushing hard if the gums are tender..
Health Tips
Gentle brushing with correct technique should be observed (not only after cleaning, but in general also).. Use a desensitizing paste like Colgate Sensitive or Sensodyne or similar if there is persistent sensitivity..