Liquifying metal and toxic fume exposure
You ask:


  Aside from burning yourself horrendously, what dangers are associated with the melting of (we can assume nearly pure) tin, particularly in a closed space? Is what he did really dangerous?


According to the SDS for tin metal from Sigma-Aldrich, tin metal (at the purity level for the product mentioned here, which is &lt;= 100% and may not be that of what was used in the video you mentioned) is not a hazardous substance or mixture. 

There is some danger specifically relating to inhalation of tin dust or aerosols (see sections 6.1 and 7.1 in the linked document):


  6.1 Personal precautions, protective equipment and emergency procedures
  Use personal protective equipment. Avoid dust formation. Avoid breathing vapours, mist or gas. Avoid breathing dust.
  
  7.1 Precautions for safe handling
  Avoid formation of dust and aerosols. Further processing of solid materials may result in the formation of combustible dusts. The potential for combustible dust formation should be taken into consideration before additional processing occurs. Provide appropriate exhaust ventilation at places where dust is formed.


In section 8.2 (Exposure controls) it is noted that:


  Respiratory protection
  Respiratory protection is not required. Where protection from nuisance levels of dusts are desired, use type N95 (US) or type P1 (EN 143) dust masks. Use respirators and components tested and approved under appropriate government standards such as NIOSH (US) or CEN (EU).


In section 11.1 (Information on toxicological effects, a subsection of Toxicological Information), information pertaining to toxicity can be found. Inhalation here will refer to tin dust or aerosols that can be inhaled:


  Acute toxicity
  
  LD50 Oral - Rat - male and female - > 2,000 mg/kg (OECD Test Guideline 423)
  
  LC50 Inhalation - Rat - male and female - 4 h - > 4.75 mg/l (OECD Test Guideline 403)
  
  Dermal: No data available


If the tin was/is at the near 100% purity level, the main risks associated with handling it are associated with the formation of dust (workplace inhalation limits are given in section 8). The HMIS (Hazardous Materials Identification System) ratings in section 16 are 1 for health hazard, 0 for flammability and physical hazard, and no value assigned for chronic health hazard.

Conclusion: Assuming the person in the video did not inhale or ingest tin dust (a proposition that largely becomes moot after it has melted, in any case), there appears to be little to no danger associated with the activity with respect to melting tin in a closed space (especially such a small amount of it).