I would like to add a little. It's some time since I read the book so I need to skip references.
@kyle-jones clearly answers for special relativity. It's effects would hardly be observable.
But there is also general relativity - which in short is all about non-flat space-time. Given the incredible gravitation on dragon's egg the effects of space-time distortion should have been notable to the cheela. Especially they should be able to see the light "bends" around the egg, and also that time flows differently in valleys and on mountain peaks. (I don't recall how high the highest mountain was, but I believe there was an chapter about this).
btw. I have never seen the equations you post as refresher outside a textbook. Professionals typically use canonical coordinates. They have the nice properties that equation of motions are the same in relativistic and non-relativistic case:
dx/dt = p/m
dp/dt = F
p and m just need to be replaced by relativistic or non-relativistic variants. And think vectors for x, p and F.
On another side not the equations of motion were mostly derived by study of planetary motion. So from that side the cheelas should come to similar equations as us.
I got also interested again. Seems my memory was wrong about the time dilation, probably mixed it up with some Interstellar alike. They did some calculations on the wikipedia talk page of the book.