To further this up now the season is over it would seem that Paddy Considine has answered this in an interview with Entertainment Weekly. In his mind Viserys had leprosy and that was what was causing his deterioration and illness.
The weight of the crown is eating away at him, as symbolized in the festering cut Viserys received from sitting the Iron Throne. By episode 3, the king had lost two fingers. That deterioration continues in episode 4 and will continue moving forward. "He's actually suffering from a form of leprosy," Considine says. "His body is deteriorating, his bones are deteriorating. He is not actually old. He's still a young man in there. He's just, unfortunately, got this thing that's taken over his body. It becomes a metaphor for being king, and the stress and strain that it puts on you, and what it does to you physically, what it does to you mentally."
Entertainment Weekly, Brothers in arms: Matt Smith and Paddy Considine unpack their 'complicated' arc on House of the Dragon
I will reiterate though, as I do below, that this would appear to mostly be a creation for the TV show and that in the books he weakens more quickly from a few wounds rather than from a disease and several wounds.
This would appear to be a slightly different retelling of the story from the books. In the show they seem to be putting emphasis on multiple cuts from the Iron Throne slowly weakening Viserys over time. The wounds shown all seem to come from cutting himself on the Iron Throne and they're just not healing all too well.
In the books all we see is one cut to his hand, but it is pretty bad:
Yet as he was descending, His Grace stumbled and reached out to right himself, and sliced his left hand open to the bone on a jagged blade protruding from the throne. Though Grand Maester Mellos washed the cut out with boiled wine and bound up the hand with strips of linen soaked in healing ointments, fever soon followed, and many feared the king might die. Only the arrival of Princess Rhaenyra from Dragonstone turned the tide, for with her came her own healer, Maester Gerardys, who acted swiftly to remove two fingers from His Grace’s hand to save his life.
King Viserys did seem to recover some of his old vigor once the new Grand Maester arrived at court. Septon Eustace tells us that this was the result of prayer, but most believed that Orwyle’s potions and tinctures were more efficacious than the leechings Mellos had preferred. But such recoveries proved short-lived, and gout, chest pains, and shortness of breath continued to trouble the king.
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This serves two main plot purposes moving forward:
- It weakens Viserys for his slightly later demise because he never really fully recovers from the wound.
- It keeps driving the wedge further between Rhaenyra and Alicent.
Essentially if the show continues this line of thought and stays mostly faithful to Fire and Blood the nature of the wounds are exactly what we think they are: just stubborn cuts from the Iron Throne.
To add onto this though from episode 5 we see an illness on Viserys that doesn't appear to be exactly the same as those from the throne wounds. Some have speculated that it is greyscale but that is pretty well known in the Seven Kingdoms and surely the Maesters would not be touching it directly or using leeching to try and heal those wounds. The show itself doesn't really do much here to explain what it is, however, Paddy Considine, the actor for the king, has stated it would appear to be a form of leprosy.
He's actually suffering from a form of leprosy. His body is deteriorating, his bones are deteriorating. He is not actually old. He's still a young man in there. He's just, unfortunately, got this thing that's taken over his body. It becomes a metaphor for being king, and the stress and strain that it puts on you, and what it does to you physically, what it does to you mentally.
Screen Rant, House Of The Dragon Star Reveals What Disease King Viserys Has