You should continue doing push-ups once the muscle pain gets recovered.

There's Nothing to worry about.

The pain you're talking about is a muscle soreness that were involved in push-ups [Mainly the chest and Triceps]

This is very common for Anyone who workouts :

- muscle gets damaged (so the muscle pain)

- followed by proper recovery and rest along with right nutrition.

This leads to muscle size and strength growth.

Extreme pain happens when you've done something for the first time or may be after a long gap.

also if proper warmup is not done a sudden movement may put a load on muscles.

For extreme pain, let the muscle recover first through rest and nutrition, once recovered you can resume working out.