If the Salesforce Connector is set to Background and Salesforce fields connected to the form are deleted or modified, will this cause any issues. Or should I disable the connector? What exactly does the disable do?
Yes, this will cause the data transfer to Salesforce to fail, as the FormAssembly system will try to send fields which no longer exist. With the connector set to background, these failures will be invisible to the end user ... you could then correct the FormAssembly Salesforce connector and attempt to resend the failed responses from the connector log view.
Disabling the connector will completely prevent any attempt to send response data to Salesforce.
If you disable the connector, there will be no entry for the responses in the connector view log, as the responses are never passed to the connector if it is disabled.
Of course the responses will still be viewable in the response tab as usual. That never changes.