I'd only ask for money if doing a very long journey, say to France, where obviously we would have agreed to split the fuel costs beforehand.
If it's just going somewhere shorter, I won't ask, but I expect that they would offer some money for petrol, or to buy food or drinks as an alternative. I myself will always offer the same.
The problem I see with saying that you don't care whether you get paid or they don't care, means that potentially, it could result in less trips. If you have a friend who always does the driving, he/she might decide they can't afford another trip somewhere due to the cost. Where as, if they knew their friends are always going to help contribute, they would be more likely to do so.