Example migrated from Codex:

The following callback removes the ‘password reminder’ link from the two main login error messages.

add_filter( 'login_errors', function( $error ) {
	global $errors;
	$err_codes = $errors->get_error_codes();

	// Invalid username.
	// Default: '<strong>ERROR</strong>: Invalid username. <a href="%s">Lost your password</a>?'
	if ( in_array( 'invalid_username', $err_codes ) ) {
		$error = '<strong>ERROR</strong>: Invalid username.';
	}

	// Incorrect password.
	// Default: '<strong>ERROR</strong>: The password you entered for the username <strong>%1$s</strong> is incorrect. <a href="%2$s">Lost your password</a>?'
	if ( in_array( 'incorrect_password', $err_codes ) ) {
		$error = '<strong>ERROR</strong>: The password you entered is incorrect.';
	}

	return $error;
} );