He suffers from Facioscapulohumeral Muscular Dystrophy:
Facioscapulohumeral muscular dystrophy (FSHMD, FSHD or
FSH)—originally named Landouzy-Dejerine2—is a usually
autosomal dominant inherited form of muscular dystrophy (MD)3 that
initially affects the skeletal muscles of the face (facio), scapula
(scapulo) and upper arms (humeral). FSHD is the third most common
genetic disease of skeletal muscle. Orpha.net lists the prevalence as
4/100,0004 while a 2014 population-based study in the Netherlands
reported a significantly higher prevalence of 12 in 100,000.
Symptoms may develop in early childhood and are usually noticeable in
the teenage years with 95% of affected individuals manifesting disease
by age 20 years. A progressive skeletal muscle weakness usually
develops in other areas of the body as well; often the weakness is
asymmetrical. Life expectancy can be threatened by respiratory
insufficiency and up to 20% of affected individuals become severely
disabled requiring use of a wheel chair or mobility scooter.
As we saw in the pilot (as described in this review), when Joe was stopped by the cop,
Ash falls from the sky, peppering the two men. “What is that?” Blake
asks. “Oh, it’s the hospital.” “Hospital?” “Yeah,” the cop says. “On
Tuesdays they burn cripples … the terminally ill. Drag on the state.”
Clearly the policy is that people with such conditions are to be euthanized.
His condition appears to have not been completely resolved as of the end of Season 1, however the actor playing Thomas Smith, Quinn Lord,
was cast in several episodes of Season 2, according to IMDB, and it was a dilemma that hadn't entirely been resolved yet, given that Smith was last seen being given the chance to do the euthanasia personally, at home, and the Nazi regime (or members within it, like the Doctor charged with reporting it) might just be sympathetic enough to give a few weeks or months of "saying goodbye time" before pushing the issue.
Updated: In season two of the show, it is revealed...
that the doctor who examined him was indeed holding off reporting the issue to anyone else out of respect for Smith, but that wouldn't keep up for long. This triggered one of the major plotlines involving Smith in season two.
Tom Smith seems to have turned himself in to medical services at the end of Season Two, out of a feeling of duty to the principles of the Reich, meaning his death is likely, although it is still not yet absolutely confirmed and last-minute attempts to save him could still be the subject of machinations in season three.