Insulin users would have better answers to this question, since they might inject multiple times a day, whereas GLP-1 users typically inject only weekly.
But in either case, the answer for subcutaneous injections using needles sized 29g and smaller is no.
Injection drug user here. We're advised to rotate injection sites but the largest issue is actually the insulin (most of it diffuses into the bloodstream but there is a local effect, usually taking the form of increased fat accumulation at sites of repeated injection), not the "making holes in the skin" part.
I don't know the pharmacokinetics of GLP-1 drugs but my guess is that they don't have the same sort of effects on SC tissue?
Before I had a CGM I did somewhere around 20,000 blood glucose tests over the course of a decade using about 1 cm^2 of forearm and the skin there is clearly not in great shape -- but it's worsened on the level of "looks like the skin of someone who is a decade older or spent too long in the sun" rather than anything medically problematic.
I’ve been diabetic for 30 years, and for more than twenty of those I did multiple daily measurements by pricking my fingertips and multiple insulin injections per day. Now I wear a sensor that I replace every two weeks and a catheter (which I change about once a week).
I really don’t understand this phobia of needles at all. After two days with one system or the other, you get used to it—there’s no pain, it’s just a mental issue of “having to make the gesture.”
My friends used to laugh at how normal it was for me to inject insulin outside a restaurant, while walking, chatting, and smoking at the same time.