Concentrated H2SO4 (over 40 %) reacts with KI according to reactions:
H2SO4 + 2KI = K2SO4 + 2HI
then H2SO4 oxidise HI to iodine
H2SO4 + 2HI = I2 + SO2 + 2H2O
When reaction cannot equilibrate one can obtain mixture of I2 and HI in different molar ratio depending on the reaction conditions, mixing, and so on.
Slighty more concentrated H2SO4 (~70 %) reacts analogously with KBr.
With concentrated H2SO4 (>= 95%) and KI you get a brown and very dirty smoke/fume of HI/I2, SO2 and H2S. I did the experiment and it really stinks. The smell of H2S is overwhelming. The reason that the fume is not purple probably is due to fuming of HI in moist air, with the I2 dissolving in the little droplets and forming H(+) and brown I3(-).
Solid very impure I2 and yellow S remains in the slurry. On dilution, the reaction only partially is reversible. The HI/I2 and SO2 part indeed are reversible, the H2S and S part are not reversible. You'll end up with a milky liquid, containing solid S when a small amount of Na2SO3 is added. Without Na2SO3 you end up with a brown turbid liquid, containing I3(-) and solid S.
The difference with KBr is quite strong. With KBr the reaction is less dirty. With KBr you get red/brown Br2 vapor and fuming HBr, but there is no H2S and solid sulphur. On dilution a yellow liquid is obtained (more SO2 than Br2 escapes, so the yellow liquid can be explained), which on addition of a pinch of Na2SO3 becomes totally clear and colorless.
PS: Please do not do this inside and if you do this, use very small amounts. As the OP already noticed, the fumes are not good at all for you!