To begin, I should say that overall the evidence is conflicting and not good with respect to your time frame (long term, "rest of their life").
Studies A, B, and C reveal animal models that show damage acutely (with recent administration of bleomycin) but also no syngery after 1 month. Of course this is an animal model. D, E, and F suggest that after bleomycin therapy (not recent administration), there is still a risk with supplemental oxygen in humans, though this is weak evidence. On the contrary, G and H suggest no correlation.
Regardless, because of some anecdotal evidence and animal model evidence, patients who have had prior exposure to supplemental oxygen may face providers who are more cautious with over-supplementing with O2 (they may target a lower, but safer, oxygen saturation percentage). They may also be more conservative in the OR when giving fluids as to not flood the lungs.
The answer is that evidence is unclear but the current practice is the above.