Sorry---no, none of the costs for driving to take care of your grandchild are deductible on your tax return. You are being a kind and generous grandparent. Perhaps the parents of the grandchild need to start helping with the cost of gas for the free child care they are receiving.
**Disclaimer: Every effort has been made to offer the most correct information possible. The poster disclaims any legal responsibility for the accuracy of the information that is contained in this post.**