Sure, a damper or shock absorber (SA for short) is like a resistor. In all types of SA energy is absorbed by friction forces and converted into heat. These friction forces are proportional to the velocity of friction elements.
That heat surely goes into the liquid or gas that is inside the SA, minus what it radiates. The amount of radiation, judging by the form factor most SA have, does not seem to be the main concern of the designer (no SA with cooling fins are popular, AFAIK).
F1 SA, I imagine, must generate low heat, compared with a normal damper. First, the "resistance" is limited, because the friction forces are limited, because the amplitude of movement is limited, because:
1. F1 cars have "hard" suspensions
2. F1 cars ride low
Second, the "current", or load, is smaller than in a normal car:
3. F1 cars weigh less than most cars
Anyway,
let's put some rough numbers to it, following a long tradition of amateurs in this site... 
The IRI (I've mentioned it
before, it's the International Roughness Index) tells you how many millimeters moves up and down the body of a car when it moves through 1 meter of road. I repost the IRI model car, Mikey_s knows it better than me, I'll say only that Ms is the body, Mus is the suspension and that xs is the movement of the body:
I suppose F1 tracks have an IRI of, let's say 2 mm/m (it could be 3, maybe, but I do not want to open the calculator, so I'll round a lot from now on).
Take in account this "quarter car" used as reference has springs, weights and SA that are different from the ones an F1 car has, which means more "travel" of the body.
Anyway, let's imagine a car on a track of 5 km with 75 laps. Without calculator I round that to 400 km. The car must move up and down 800 m through the entire race (at 2 mm per meter of track).
Of course, this is the "throughtput" of energy through the resistor, not the energy damped (on the contrary, it represents the energy NOT damped, but let's assume they are of the same order of magnitude). With a gravity of 10 m/s2 I get:
Energy = m*g*h = 600 kg * 10 m/s2 * 800 m = 5 Mjoules
If the race takes 1.5 hours, which is like 5.000 seconds, give or take, you have:
Power = 5.000.000 Joules / 5.000 seconds = 1 Kwatt
This is more like 500 watts if you take in account that half of the energy (at least, if no more) goes to the rear shock absorbers.
Now, there are other things, beside SA, that adsorb energy, that is, wheels and springs, shown on the posted picture as "kt" and "ks". Let's assume that wheels and springs dissipate only half of the energy, even if that's probably conservative for an F1 car, famous for relying on wheels to provide shock absorbing power.
We arrive to a power around 250 watts, which is like three light bulbs. That's enough to keep your legs a tad warmer than my taste. Anyway, part of this heat surely can be managed by simple cooling systems.
As far as I remember, there have been posted in this site some pictures showing air vents on top of the frontal part of the cockpit, surely for cooling. I suppose the energy of the body inside a fire suit is larger than the numbers I got.
Surely I must have made some mistakes, I appreciate if somebody has the time or the interest to point them out.
). It's going to be limited to 12.500 Newton or, let's say, 1.2 tons. That'll double my numbers. You could cut them in half again if you think there are two dampers, so the numbers are valid for one damper. As I write that I think something must be wrong... I remember that figures of downforce arrive to several times the weight of the car.