Would they move to the next in line anyway, or would they give the heir special arrangements? Were they barred from inheritance as soon as the disability was discovered or caused? And were some disabilities better or worse received than others? I specifically mean physical disabilities, not mental ones.
Depends on place and time. For example in Byzantine culture the Emperor was considered a reflection of heavenly authority as the representative of the God to rule the Empire.
If the heir had a disability he was consider incapable of taking the throne and someone else became the Emperor. A famous exception was Justinion II the slit-nosed "Ρινότμητος" who had he nose cut of and overthrown but was able to become an Emperor again and had a golden replica of his nose attached to his face. Another example was Michael IV the Paphlagonian who was epileptic but kept the throne until he died after a ling illness.
Byzantine Emperors also used to blind their rival as a blind man was also incapable of leading an army into a battle, something that was very common for taking control of the Empire. Lastly castration was also very common as a castrated man cannot produce heirs that could challenge the Emperor or his children for the throne.