I don't think it would. In a zombie (or other type) apocalypse dollars have zero value. You can't eat them or use them as fuel. Gold may have value as a medium of exchange but it would quickly revert to barter system for food, fuel etc.
What good is selling food for dollars if you can't spend those dollars next week? Better to swap the food for another good which can be used or exchanged.
Just existing doesn't give them value. After an apocalypse event, even with some sort of US military and US government functioning dollars will lose their fungibility quickly. What could happen is an increased supply of them e.g unprotected banks/vaults could be robbed. But this would have the effect of making more dollars available, so although they exist they are not scarce, if they are not scarce then why would I swap them for 10gallons of fuel? ATMs would be easily looted and are widely located.
What good is selling food for dollars if you can't spend those dollars next week? Better to swap the food for another good which can be used or exchanged.