Does going faster always result in more calories burnt

All other things being equal, yes. The increase in effort more than makes up for the decrease in ride time.

Consider two scenarios where one rides 20 km: first at 25 km/h, second at 30 km/h. In the first case, riding at 25 km/h takes 96 W of power, in the second, riding at 30 km/h takes 150 W according to this web page (using my own stats).

 96 W * 20 km / 25 km/h = 276 kJ over 48 minutes  
150 W * 20 km / 30 km/h = 360 kJ over 40 minutes

Thanks to a lucky mathematical coincidence, kilojoules of work is almost exactly equal to calories burned. Some estimating tools treat calories burned as 10% above kJ of work.

Speaking purely in calories and power, this is just a small elaboration on Adam's answer: the power required to maintain a certain speed against aerodynamic drag is proportionate to the cube of that speed. The time to complete a distance is inversely proportional to speed. Thus, you will always burn more total calories if you increase your speed. The fact I asserted about power and aerodynamic drag is available in the calculator that Adam linked.

I am oversimplifying a bit, since I'm ignoring elevation change and rolling resistance. The power to overcome rolling resistance (that is the power consumed to overcome the friction between tires and ground) is linearly proportional to your speed. However, I believe that the speed where aerodynamic drag is greater than rolling resistance is quite low, possibly in the region of 10mph. Whatever the case, the total power required to hold a certain speed does not increase linearly, thus giving you your answer.