i have a 96 ford f-150 4x4 and on the back i am running the right size tires but they are car tires ! according to the sticker on the door it takes 235-75-r-15 XL tires ! the front is already that size but would it handle any better on the road if i took the car tires off and run truck tires on the back ? i bought the car tires because they were cheap and i needed 2 tires at the time !
If you're just driving on the street and carrying groceries you should be fine. Where you will have trouble is if you try going off-road or carrying any heavy loads. The tires you have are made for a car. Truck tires are built to take weight and some abuse. If you don't use it like a truck just keep driving.
The one problem with the car tires is that they are not designed to handle the weight of the truck and a load. They have a lower load range rating. Depending on the car tires and the number of ply rating they might also flex a bit more than the truck tires which means a little more movement in the rear so not quite as good of handling. You may not notice a handling problem until you place a decent load in the bed of the truck. Then the car tires will show their weakness compared to the proper tires. Another problem is under a load the car tires may heat up more and this could lead to a failure of the tire.