I have an automatic chicken cooker. However, the grow-up chickens seem to not immediately take damage from lava when it is dispensed from a dispenser, they just sit there, heads in lava and not a single point of damage do they take, at least for half a minute or so.
I do have a half-slab resting on the block where the chickens rest,so the grow-up chickens should be taking damage...

However, they do take damage from lava manually placed.
Below you can see a shot of the chicken's head clearly in lava, but no damage being taken.