Hi Amy,
The eye disease operates separately, and so it is not believed that any particular treatment for the thyroid disease can affect our experience with eye disease.
You should know that eye involvement in Graves’ patients who are diagnosed in their youth is much less common, and typically involves the fatty tissue surrounding the eyes (instead of the muscle tissue). Fatty tissue involvement resolves much better, if it happens.
In addition, those who suffer the worst of the eye disease are a small percentage of the total Graves’ patient population, so usually it’s not something you need to actively worry about.
I should point out that one study of post-RAI patients found "temporary worsening" of symptoms following RAI. Neither term was well defined, and the population affected was about 16% of the patients. That number was reduced to 0% when the patients took oral steroids for a short period of time (a week or two) prior to and following the RAI dose. Steroids carry significant risks, even a short course like this, so usually doctors opt against the steroids unless eye involvement is already moderate or severe. It’s something to discuss with an ophthalmologist if you’re considering RAI treatment.