A clogged toilet or a clogged vent pipe should not directly affect your dishwasher. The plumbing systems for toilets and dishwashers are typically separate, and their drainage systems should not be interconnected.
However, it's important to note that both toilets and dishwashers require proper drainage and venting for optimal performance. If you have a severe plumbing issue, such as a blockage in the main drain line or vent pipe, it can potentially impact the overall drainage system of your house. In such cases, it's possible that multiple fixtures in your home, including toilets, sinks, and dishwashers, may experience drainage problems.
If you're experiencing drainage issues with your dishwasher along with a clogged toilet or a vent pipe problem, it could be an indication of a larger plumbing issue. It's recommended to have a professional plumber inspect your plumbing system to identify and address the underlying problem. They can determine if there are any blockages or venting issues that are affecting the proper functioning of your dishwasher and other fixtures.