If Andrew Cuomo fills Hillary Clinton's Senate seat, how will this impact Chris Cuomo's job at ABC News?

No this should not be a problem. Chris would never be able to do a report about Andrew Cuomo but he could still anchor a story about him. That would be weird for him to anchor that but he could still do it. He probably would be the replacement for Diane if she left but I don't think she is leaving. I hope she is not at least. I would be so mad if Bill Weir replaced Diane if she left.