Temperature Converter

Python 3, 118 116 bytes

I=input()
t=eval(I[:-1])
u=ord(I[-1])%7
exec("u=-~u%3;t=[t*1.8-459.67,(t-32)/1.8,t+273.15][u];print(t,'FCK'[u]);"*2)

Performs the conversions in a rotary order K -> F -> C -> K twice.

CJam, 77 65 60 59 55 54 52 bytes

l~2|"459.67+1.8/'K 273.15-'C 1.8*32+'F"S/m<{~N2$}%6<

Try it online in the CJam interpreter.

How it works

l~    e# Read and evaluate the input: F, K, C -> 15, 20, 12
2|    e# Bitwise OR with 2: F, K, C -> 15, 22, 14 = 0, 1, 2 (mod 3)

"459.67+1.8/'K 273.15-'C 1.8*32+'F"S/

      e# Push ["459.67+1.8/K" "273.15-C" "1.8*32+F"].
      e# These commands convert from F to K, K to C and C to F.

m<    e# Rotate the array of commands by 15, 22 or 14 units to the left.
{     e# For each command:
  ~   e#     Execute it.
  N   e#     Push a linefeed.
  2$  e#     Copy the temperature for the next iteration.
}%    e# Collect all results in an array.
6<    e# Keep only the first 8 elements.

JavaScript (ES6), 127 130 132 bytes

Waiting for some amazing esolanguage answer, I did not find much to golf here.

Using template string, the 3 newlines are significant and counted.

Run snippet in Firefox to test.

F=x=>([t,u]=x.split(' '),z=273.15,y=9/5,u<'F'?`${t*y+32} F
${+t+z} K`:u<'K'?`${t=(t-32)/y} C
${t+z} K`:`${t-=z} C
${t*y+32} F`)

// Predefined tests

;['23 C','86.987 F','56.99999999 K']
.forEach(v=>O.innerHTML += v+' ->\n'+F(v)+'\n\n')

<input id=I><button onclick='O.innerHTML=F(I.value)'>-></button><br>
<pre id=O></pre>