Does the brain consume more calories when we think harder?

Yes the brain does consume more calories when "thinking harder" (more active or engaged).

When your brain is more active, compared to a rest/slow wave sleep state, more of your neurons are firing, and they are firing more often.  When firing they release neurotransmitter (along with several other types of messengers) which then need to be replenished by the neuron.  Creating more neurotransmitters takes energy.  So the more neurotransmitters being released, the more that need to be created, the more calories you are burning.

Interestingly, on average children's brains consume two times as many calories as adult brains do. [1,2]

[1] http://www.dana.org/news/cerebrum/detail.aspx?id=1228

[2] http://www.med.wayne.edu/wayne%20medicine/wm97/brain.htm

This is an answer I wrote on Quora.