If two binary brown dwarfs began to exchange gas with each other, would they both ignite into stars?
It is possible for two brown dwarfs to form a contact binary. This would not, of itself, cause core temperatures in either to rise. 

If the two brown dwarfs were to merge, the mass of the resulting body could be enough to generate sufficient heat and pressure in the core for nuclear reactions to start. If both brown dwarfs had a mass of about 60 Jupiters, the combined size of 120 Jupiters would certainly be big enough to form a red dwarf.