Implications of Violation of Independence Assumption in ML Models

This is a somewhat broad question, but I'm having trouble finding a good answer anywhere. I know many ML models will impose an independence assumption in the data. But I'm having a hard time really understanding what the practical implications are if that independence assumption is violated. For linear/logistic regression, I get that it likely biases interpretation of the coefficients. But what about from a predictive performance standpoint? Does violating the independence assumption actually matter?