Will anything bad happen if I ignore tendinitis-like injuries?

I have what I am pretty sure is a tendinitis-like injury. This is the third time this has happened (all involving different tendons in my feet!)

I understand this is an overuse injury and I should just take it easy. But hypothetically, if I just ignored the pain and kept running on it, am I really risking any more serious damage, or am I just prolonging the pain?