Exercise is never a bad idea, unless you're nursing an injury or fighting an infection.
In the case of tinnitus and hyperacusis, as you have identified, it is simply a matter of which exercise you choose to do, to avoid worsening.
I suppose "heavy" is a relative term, but yes, I understand lifting free-weights
contributed to the deterioration of
@Brian Newman's condition.
I lift
light free weights, but heavier machine weights, and haven't had any problems (touch wood) to date.
This is the exact protocol I adopted to return to exercise after developing severe tinnitus and hyperacusis in 2009: Push-ups, Sit-ups and running around my local park.
The
likelihood is that your tinnitus will be
temporarily elevated after the workout.
You may also experience pulsatile tinnitus during and/or after the workout (for a couple of hours). I know of people who both have and don't have tinnitus, who encounter pulsatile tinnitus during and after workouts; it doesn't seem that uncommon, especially in those of us with compromised hearing.
In any case, none of these exercises alone will do you any harm.
The only thing I would advise against is the compulsion to wear earplugs during the running for example.
Obviously as a severe hyperacusis sufferer, you're more likely to fixate on the sound of your own footfall, or become paranoid about the reverb travelling from your legs to your head, but trust me, the occlusion with earplugs in will be much worse.
If by some small chance you
do find the sound of your own running too disturbing, you can always invest in an exercise bike, which would be fine to use with earplugs in, due to the lack of any vibrational contact with the ground.
For the record (as a moderate hyperacusis sufferer), I use both a cross-trainer at the gym (with earplugs), and an exercise bike at home (without earplugs) and both have been absolutely fine... except for the
temporary elevated and pulsatile tinnitus I mentioned before, of course.