My go-to strength training workout used to be 100 overhand (palms facing away from me) pull ups on a bar. I did this one day a week. I couldn't do them all in a single set, of course, but I would do small sets of 3 to 6 over the course of an hour until I hit 100.
The 100 pullup workout was amazing for building strength. It would leave me sore as hell but it quickly produced large, visible gains in my my arms, trapezoids, lats and other back muscles.
Eventually the pump party came to an end when I got a tendonitis in the top of my palm where the bar pressed the hardest into my hand. It started to hurt slightly and i just solidered on through and ignored it: until one day it flared up bad. it It was very painful for weeks (and hurt for months) and it serious curtailed my ability to grip any kind of weight or bar and do most kinds of workouts. It was a major setback and that sent my fitness backwards.
Your muscles are stronger than your joints, in some cases. It's important to do varied sets of different exercises and limit the intensity and overuse of one particular body part. There is a reason why sets of 8 to 15 are popular among bodybuildiers and trainers, as it represents exertion but not over-exertion with too much weight or too many reps. Overtraining can lead to injury or too much cortisol / stress on the body for optimal gains.
I don't think the 100 pushups a day challenge is unreasonable or bad for healthy younger people to do. If I were doing it myself, I would do it 2 or 3x per week (M-W-F) to give my muscles time to recover. Pushups are less stressful on the body than pull ups but they are still strong full-bodyweight exercise.
I recommend doing pushups on an empty stomach (first thing in the morning, before dinner, before beds, etc.) Doing pushups with a recent meal in your stomach can cause acid reflux.
Use excellent form: tighten your core, straighten your body, keep your elbows tucked "in" more close to the body rather than flared out. If you can't do a clean, strong, solid pushup anymore - stop and resume later rather than shaking and tweaking your way through several more reps.
If you are feeling sore or you are new to working out - listen to your body and back off. You don't have to prove anything to anyone on the internet - just do your personal best and stay strong for the long haul rather than burning out or hurting yourself.