For Federal income tax purposes, whether proceeds are taxable depends on whether you have a gain (selling price is greater than selling expenses and price paid) or a loss (selling price is less than the vehicle purchase price plus selling expenses).
Personal losses are not deductible [IRC § 165(c)(3)]; gains, however, most definitely are. Sales of most personal automobiles will not have any effect because most cars are depreciating assets. However, in a few cases (antique cars, luxury cars) there could be a taxable gain on sale.
I don't know offhand if any states with personal income taxes differ from Federal rules regarding personal losses.
Keep in mind that if your state imposes sales/use tax the buyer of the automobile will be assessed use tax when he goes to register the vehicle. This is definitely a consideration for the buyer.