Remove leading and trailing zeroes

Jelly, 2 bytes

Code:

t0

Explanation:

t   # Trim off...
 0  #  zero at both sides

Try it online!

JavaScript (ES6) 43

a=>(f=a=>a.reverse().filter(x=>a|=x))(f(a))

Less golfed

a=>{
  f=a=>a.reverse().filter(x=>a|=x) // reverse and remove leading 0
  // leverage js cast rules: operator | cast operands to integer
  // an array casted to integer is 0 unless the array is made of
  // a single integer value (that is ok for me in this case)
  return f(f(a)) // apply 2 times
}

Test

F=a=>(f=a=>a.reverse().filter(x=>a|=x))(f(a))

function test(){
  var l=(I.value.match(/\d+/g)||[]).map(x=>+x)
  O.textContent=F(l)
}

test()

#I { width:90%}

<input id=I oninput='test()' value='0 0 1 3 7 11 0 8 23 0 0 0'>
<pre id=O></pre>

CJam, 13 bytes

l~{_{}#>W%}2*

With the array inputted.

Longer version:

l~             Puts input on the stack and parse as array
  {       }    Code block
   _           Duplicate the first thing on the stack
    {}#        Finds the index of the first non-0 value in the array, puts it on the stack
       >       Slices the array from that index
        W%     Reverses the array
           2*  Does the code block twice in total