Detriplicate a string

Jelly, 6 bytes

Qx2œ&@

Try it online! or verify all test cases.

How it works

Qx2œ&@  Main link. Argument: s (string)

Q       Unique; deduplicate s.
 x2     Repeat each character.
   œ&@  Take the multiset intersection of s and the previous result.

JavaScript (ES6), 42 48

Edit A whopping 6 bytes saved thx @Neil

s=>s.replace(k=/./g,c=>(k[c]+=c)[11]?'':c)

Explanation: I use the properties 'a'...'z' of object k to store info for each character (object k is a regexp in this case just to save bytes). These properties are initially undefined. In javascript adding a number to undefined gives NaN (quite sensible), but adding a string 'X' gives "undefinedX" - a string of length 10 (silly). Adding more characters you get longer strings. If the obtained string for a given character is longer than 11, that character is not copied to output.

Test

F=
s=>s.replace(k=/./g,c=>(k[c]+=c)[11]?'':c)

test=`

xxxxx
xx
abcabc
abcabc
abcdabcaba
abcdabc
abacbadcba
abacbdc
aaabcbccdbabdcd
aabcbcdd`.split`\n`
for(i=0;i<test.length;)
  a=test[i++],b=test[i++],r=F(a),
  console.log(r==b?'OK':'KO',a,'->',r,b)

Python 2, 48 bytes

lambda s:reduce(lambda r,c:r+c*(r.count(c)<2),s)

c[r.count(c)/2:] is a same-length alternative to c*(r.count(c)<2) .

49 bytes:

r=''
for c in input():r+=c*(r.count(c)<2)
print r