No, you're not obligated.
But nor do I like PaulInPerth's example. Cab drivers get paid for gas, wear and tear on their car, depreciation, labor/profit/opportunity cost, extra licensing/insurance, and also the extra risks they take. It's totally different than a friend doing you a favor.
[For a shorter answer, please skip to the last paragraph.]
And I also disagree with Paul about chipping in for gas now. While offering to pay for gas is usually a good idea (more gas than actually needed), I would not retroactively chip in for the gas in this case either, the timing of that gesture, now that you already know he got a ticket for helping you, would send the wrong message.
In addition to the gas, time out of his day, and the inconvenience, he took an extra risk by doing you that favor (and perhaps saved you from asking someone else or saved you from renting a moving truck). That's really what you owe him for, that intangible, not the fact that he got a ticket because of a mistake he made.
Plus, there is also the cold hard calculated fact that you might need him to help you with the same thing in the distant future (and he might feel hesitant about that, considering this experience and the fact that his skills as a driver will probably not have improved much by then). And there is also the fact that the few city people I know who own trucks, vans, SUVs, or cars large enough to carry sofas, usually get asked to help others move all the time. And by that, I really do mean all the time, there is even a bumper sticker for it.
And finally, there is also the income differential. I know this shouldn't matter. But since this brother-in-law is essentially part of your family now, it could. If either he or you are currently unemployed, that traffic violation could prove very expensive.
But assuming that the income of both your households are ok enough and that this traffic ticket won't break the bank with either of you, this is what I would suggest. Don't pay him. But do arrange to pay him back more subtly in the very near future with a nice dinner for the 3 or 4 of you, or an in-kind gift he might really appreciate, or some other gesture of value that he can't refuse.