Fuel dumping seems to have a lot of restrictions, like the need to do it above a certain altitude and over water. It also has to end up somewhere, even if it vaporizes, which is less than desirable. It seems like burning off fuel as it's dumped could solve these problems. Obviously there's the risk of rapid unplanned incineration, but what are the other factors?
There's no benefit to burning dumped fuel, it actually introduces risk to the airplane which is dumping fuel and any other aircraft in the vicinity and possibly people on the ground depending on the altitude at which is done. Building a system to burn it is not worth the time, effort and money given there's no good reason to do it.
Dumped fuel will vaporize and mix in with the air, and then break down over time from exposure to air, heat and UV light. This is not ideal for sure, but burning it isn't environmentally friendly either as it produces CO2, CO and soot. Evaporation of dumped fuel is a miniscule problem when looked at in context as it's such a rare occurrence. Canada measured 58 million liters of gasoline and diesel lost to evaporation from retail outlets and pump in 2009, if you extrapolate that to the US it's got to be over 100 million US gallons of fuel evaporating into the atmosphere every year. An airbus A380 contains 320,000 liters of fuel, if you dumped all that at once it's not going to make a statistical difference.
Obviously there's the risk of rapid unplanned incineration, but what are the other factors?
"Rapid unplanned incineration", although appearing nowhere in the certification standards for any airplane, will be fairly high up on the list of thing aircraft designers try to avoid. The linked F-111 dump-and-burn is very much an exception that is highly unlikely to occur in any civilian aircraft.
Dumping fuel from higher altitudes has not yet produced a problem for anyone, burning it as it leaves will require extensive (and expensive) testing that no one is interested in. Save it for the air shows.
As GdD explained, while aircraft are supposed to be careful about where they dump fuel, the dumped fuel is not a significant hazard to health or the environment.
Burning the fuel represents much more risk than just "rapid unplanned incineration". Large airliners departing for a long flight can be carrying enough fuel to fly for 6 to 10 hours, and must dump this fuel in less than a few hours. All of the energy that would have gone into propelling the aircraft over a long period of time is released as heat in a much shorter amount of time. This heat will be a problem.
Fuel dump nozzles are typically placed on the wings. This allows the fuel to be dumped clear of the fuselage and engines, and is conveniently located right next to the fuel tanks in the wings. If the fuel is burned directly from these nozzles, it will cause some issues. The nozzles will have to be designed to withstand the heat. The wings in this area also need to be designed for this heat, and prevent the fuel (vapors) inside from igniting. The passengers would probably not be thrilled to see giant flames coming from the wings, and would definitely feel the heat.
The fuel dump nozzles could be moved but this adds complexity and risk. The other logical place would be near the APU in the tail of the aircraft. The tail would still be affected by the heat, and in some large aircraft there are still fuel tanks in the tail that must be dealt with.
All of these challenges, on top of controlling a massive fire outside of the aircraft, and burning the fuel provides relatively little benefit. Simply dumping unused fuel is much safer for everyone involved.