I have never crewed on a helicopter that dumps fuel, nor have I known of any machine in the fleet that even has that capability - though I have been wrong before.
It has been noted already that helos can take off and land at max Gross Take Off Weight (GTOW), or all up weight (AUW) and thus do not require weight shedding to land safely (as we don't have wings to store fuel so we don't have to worry about over-stressing them on landing). However, keep this in mind: different fuel loads alter the emergency course of action in the case of a forced landing.
When considering helicopter performance (and we will talk in terms of torque), we explore 2 main ideas - power required and power available.
Power required is the amount of torque that I need on the main rotor to overcome gravity and get into hover flight. Hover flight is always the most power hungry because i'm drawing air down into the rotor in order to create the smooth fast airflow over the blades and then pushing it down to the ground. There are two modes of hover flight - Hover in ground effect (HIGE) and hover out of ground effect (HOGE).
Hover In Ground Effect
Hover in ground effect is where you are close enough to the ground (around 4 feet) that you draw the air down through the rotor disk and it pushes in into the ground. In accordance with Newton's Third Law, as the downwash pushed into the ground, it also pushes back towards the rotor, creating a high pressure "bubble" beneath the helo, and reduces the recirculating air on the rotor tips.
Hover Out of Ground Effect
Hover out of ground effect is where we are roughly over a rotor span in the air - say around 50 feet. We have now lost that cushion of air beneath us because there is no air trapped between us and the ground. It simply recirculates around and around.
Sometimes a picture is worth a 1000 words:
As I hover in ground effect, I get a bit of help because I have that bubble helping keep me in the air, meaning my rotor has to do less work (really its less AOA or pitch pulled from my collective), whereas I don't have that help out of ground effect. The difference (at least for the bell 412) is around 11 percent torque. This means at a given take off weight, I will require 11 percent less torque in ground effect than out of ground effect to maintain hover flight.
Getting back to power required, this is the torque needed to lift the sum of all the weight on board my helo (and the helo itself). This means the more cargo + more fat bodies + more gas in the tanks = more power required. This is going to affect my HIGE and HOGE torques, along with my minimum safe airspeed (which i'll cover later).
Power available is the second half - this is what the engines and rotor can produce in terms of power and thrust/lift. The 2 main factors are altitude and temperature. We measure these in terms of torque as well - mast torque for the rotor and engine torque for the motors - but for now we are only going to focus on mast torque. Essentially, this is the amount of final drive power we have in our pocket. Air is power - so the more air we have, the more torque we produce. Cold air at sea level in the Arctic will produce more power than hot air in the mountains of Afghanistan - see density altitude if you want to learn more. For a given altitude and temp, I will be able to produce a certain amount of power measured by rotor torque percent. A cold day I could get 100% mast torque. A hot day I might only get 85% mast torque because I've run out of engine turbine temp to produce power or I've run out of collective pitch because my rotor blades are at max pitch. Most helicopters are also multi-engine, so two engines running will get me 100% torque, where if I lose an engine I might only get 60% mast torque.
Finally there is a safe minimum airspeed for single engine operations (there is the deadman curve for twin engine, but we are talking emergencies here). As a helo hovers, it draws air down from above to create the airflow over the blades - essentially the spinning of the rotor is solely responsible for creating relative wind. As we start to translate into forward flight, airflow is established over the blades by forward movement, creating a relative wind more in line with the blade chord. Our forward flight means the helicopter depends less on pure blade rotation to create air movement for lift because forward flight is providing the clean relative wind over the blades. This lets us stay in the air with less torque required.
If you made it this far, great!
So how does this tie all together? Lets use an example:
All up weight: 11,500 lbs.
Fuel on board: 2000 lbs.
Mast torque available (twin engine): 100%.
Mast torque available (one engine inoperative - 2.5 minute power): 65%.
Mast torque available (one engine inoperative - continuous power): 54%.
Minimum airspeed (single engine): 25 kts.
Hover torque required (in ground effect) 75%.
Hover torque required (out of ground effect) 86%.
Helicopters landing at or near maximum all up weight is normally limited to a factor of power required/available rather than a factor of airframe loading limits/overstressing
You are in ground effect hover and lose an engine. You pull the collective into your armpit and accept settling with power because your power available single engine (65%) cannot meet the power required (HIGE 75%). Fuel dumping is not an issue here.
You are out of ground effect hover and lose an engine. Again, you pull the collective into your armpit and accept significant settling with power because - you guessed it - power available single engine (65%) cannot meet the power required (HOGE 86%) and you accelerate to the earth. The whole iteration takes about 5-7 seconds, and the result lands you as a flaming fireball on the 6 oclock news - however you don't get to see it. This is why you don't hover heavy OGE, and why you don't violate your deadmans curve! No time for fuel dumping, so not an issue here.
You are in a high hover (HOGE) - say 150 feet. You lose an engine, but instead of pulling the collective into your armpit and dying, you pull some power and push the cyclic forward - accelerating above safe single engine speed (25 kts). You start to arrest your descent as you trade altitude for airspeed, bringing enough speed in to climb out safely. You can then either choose to stay in the air on single engine for the next hour to burn enough fuel to facilitate a HIGE landing, or you can carry out a run-on landing at or above 25 kts at your current weight. No fuel dumping required.
You do a no-hover (or rolling) takeoff and climb out above minimum safe single engine speed (25 kts), and then lose an engine. You have two options - continue the climb out and burn some gas until you bring your all up weight down to a torque that will allow you to hover single engine 2.5 minutes (which will take at least an hour - thats why flight engineers bring ipads and magazines), or simply continue your circuit and carry out a run on landing (or rolling landing) at or above 25 kts. Nice, slow, not a lot of runway needed. Fuel dumping not an issue. Keep in mind that a nice slow decent with loaded blades is key - every approach is an approach to overshoot with the option to land.
Remember, the fuel loads in most helicopters - even the big ones - are measured in thousands of lbs and not tens of thousands of lbs. We don't carry a significant amount of gas to really alter our landing profile - all we need is enough to stay in the air for a couple of hours. After that, our backs, necks and egos are sore from terrain flight and vibrations.
As an extra note, trying to dump fuel in the hover would be a hazardous mistake. As you dump fuel mist, it would recirculate through the rotorwash back onto the helo - into the engines and all over the place in general. Then after it lights off, your land as soon as practical emergency just turned into a one bell alarm. IF it were to happen, you would need an established minimum airspeed to carry it out, but then you certainly wouldn't be transferring it to a ground storage tank.