Is one operation on two separate things singular or plural?
The correct option in this sentence is "is".
"Burning fossil fuels like [oil and gas] is the main source of greenhouse gas emissions from human activities". 'Is' refers to 'the process of burning of fossil fuels', which is singular.
If the sentence were "burning and [another process] of fossil fuels like [oil and gas] are the main sources of greenhouse gas emission from human activities", you would use are because 'burning and [another process]' is plural.