First, some back-of-the-envelope assumptions. Take these with a grain of salt!
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Install penetration: Flash claims 97% install penetration on the market.
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Timespan: Flash was introduced in 1996. Let's assume it didn't hit that market penetration until 2002, and we'll ignore all the years before that.
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Internet user base: The total number of people that can reach the Internet right now is about 2.25 billion. However, it only had about 350 million people in 2000. So let's say that on average the Internet had 1 billion people using it over the timespan in question.
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Power usage: Typical desktop quad-core CPU power usage at maximum utilization: 70 to 100 W. Let's be conservative and say that at most, 25 W of that power is dissipated because of Flash (since at least some of it will be consumed by browsers, the OS, other programs, etc.).
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Flash runtime fraction: Let's assume that the average Internet user's browser is running a Flash plugin process 5% of its operating time. Let's also assume that users actively operate Internet-connected devices for 6 device-hours per person-day on average. So Flash is running 25% * 5% = 1.25% of the time.
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Ignoring additional power usage: Let's ignore the extra power used by additional memory (e.g. if people had to buy more memory just to run Flash, we won't count the power drain of those extra modules), onboard graphics cards, and hard drives (e.g. if the disk needed to swap a lot while running flash).
Now for the math. Let's substitute our numbers:
energy consumption = power per device * # 25 W/device
number of people * # 1 * 10^9 people
runtime fraction * # 0.25 devices/person * 0.5 = 0.0125 devices/person
install penetration ratio * # 0.97
timespan * # 10 years
So we wind up with:
energy consumption = 3.031 billion W-yr
= 26.55 billion kW-hr
For reference, that's about the energy produced by all the world's nuclear power plants over three days, or about the energy produced by the sun's rays striking the earth for half a second.