It depends entirely on 2 things; (1) what you're burning and (2) what you do with the heat liberated. Burning plastic on an open bonfire is not good. Burning the same plastic in a well-designed, high-temperature furnace with a heat exchanger supplying steam to a turbine to generate electricity, and dissipating the waste heat through a neighbourhood heating scheme would be much better since it ultimately means that less fossil fuel is being used for heating and power. Small, distributed CHP schemes also reduce the amount of fuel used in transporting waste.
I'm assuming you mean burning stuff in your garden on a bonfire instead of putting it in a bin to be collected.
Burning 'man made' materials (particularly plastics) on a bonfire would not be good, you would be releasing all sorts of gases, poisons, particulates, CFCs, toxic vapours, cyanide etc.
As far as CO2 and other green house gas emissions is concerned burning vegetation won't add any new carbon to the eco system, however there is a problem of particulate pollution and dioxins etc.
However, there is a catch in this issue. By burning wood and other vegetation you are taking the carbon from the wood and pushing it into the atmosphere as a green house gas. The logical conclusion is that if you continually burn vegetation and don't replace it you will create an imbalance with to much CO2 and other GHGs in the atmosphere, because you no longer have enough trees to take the carbon back out of the atmosphere.
So although burning vegetation doesn't upset the general balance, doing it to much without planting replacement vegetation could be just as bad over a short term as burning fossil fuels. If you burn vegetation (often wood) it must be sustainable.
I mean 'Vegetation' in a broad sense and would include wood, paper, cardboard, branches, leaves.
