How does doing heavy computation affect the power consumption and heat generation of a CPU?

Modern computers use significant power only when doing work. So if you're doing more work, you're drawing more power. A properly designed PC is able to deliver all the power it'll ever need, but in practice may rarely or even never actually get there. At the high level, there's software power management. Many devices can be shut down when not being used.. smart I/O and system software does exactly that. This is kind of hit or miss on PCs, better on laptops, and generally very good on mobile devices. And of course, even if something's running, if it only has a little work to do, it can go into a low-power idle state for much of the time. At the low level, it's all about CMOS transistors. CMOS means "complementary Metal-Oxide Semiconductor", which basically says that for any switch, you have one transistor to switch a signal to "1", another to switch it to "0" -- that's the "complementary" part. A CMOS transistor is basically a little wire made of silicon that can be turned off or on via a voltage applied to a "gate". The transistor is an excellent switch -- when off, it presents a nearly perfect open circuit, when on, a conducting path with a resistance in the milli-ohm range. The problem is switching -- when you change the state on a CMOS gate, both transistors transition through the "active" part of their operation, in which that little silicon wire acts like a resistor. A resistor is basically a device that converts power to heat. So as you switch CMOS devices, they consume power... when sitting still, barely any.

CMOS gates only draw power when they change state, so all things being equal, the CPU draws more power when it's doing something computationally intensive than when you're just surfing the Web. I can give anecdotal proof of this. My computer occasionally shuts itself off when the BIOS senses the CPU temperature is too high. This happens only under one set of circumstances - it's a hot day, the CPU heat sink has got a bit clogged with dust, and it's doing a video format conversion, for example converting an MP4 movie to XVID. This  format conversion is probably the most computationally intensive job my computer ever has to do, involving heavy use of the maths coprocessor. That's a second processor-within-a-processor which, in normal activity, never gets used at all.