Can doing pushups irritate tennis elbow?

Yes: Lateral epicondylitis or tennis elbow can be aggravated by any activity that places the wrist in extension. Depending on the severity of the condition, one may be able to continue to do pushups. There is great variability in the severity of the condition and the activities that aggravate it.

Yes: The pain from tennis elbow comes from when the elbow is fully extended. However, any stress on the elbow can cause it to become inflamed. Thus is generally tell my patients, if it hurts -- stop, if its sore -- slow down. Listen to your body.