You can do everything except shoot. This includes Dash, Jump, Melee, Parry, Hero Abilities and Active Items. After some testing I've confirmed this bug works with the following heroes:
Bebop - Use abilities during ult.
Dynamo - Dash, Jump, Melee, Abilities, etc. during ult, very broken.
Grey Talon - Abilities during ult and charging shot.
Lash - Abilities during ult.
Mirage - Abilities, dash, jump, etc. during ult.
Mo and Krill - Can ult during burrow.
Pocket - Can go into case while ulting.
Seven - Abilities during ult.
Sinclair - Can copy bugged abilities.
Warden - Dash, Jump, Abilities during channel, very broken.
Yamato - Dash and jump during ult channel.