TapeEquilibrium implementation does not satisfy all requirements
I coded up my own solution, and mine was pretty close to yours, but mine received 100%. 

Here are the problems: 

List&lt;int&gt; list = A.Cast&lt;int&gt;().ToList();


Both the Cast and the ToList are redundant here. Arrays implement IEnumerable, which is enough for LINQ. The ToList is iterating the entire array, which is costing you some runtime.

int temp = 0;
int sum = list.Sum();
List&lt;int&gt; result = new List&lt;int&gt;();
foreach (int i in list)
{
    temp = temp + i;           
    result.Add(Math.Abs(  ( sum - temp) - temp)); 
}
return  result.Min();


This is hurting you on the space requirements -- you're generating a list of every subtraction. Then it's hurting you on time requirement again because it has to iterate the entire result list to find the lowest number.

You can do the entire thing with two passes of the input. You're doing 4. 

I'll also say that you should almost never have a variable named temp. Give it a name that describes what it actually is. I used the names "left" and "right". 

Left starts at 0, Right starts at the Sum. For each number, add it to Left, subtract it from Right. Much easier to read.

Update
I actually realized that in addition to everything I said, there's a major logic flaw in your version, which is probably what's hurting you more than the other stuff:

What happens if your code tries to handle {3, -1}?

First pass:

temp = 3
sum = 2

|(2 - 3) - 3| = 4

There should be no second pass, but you do one anyway:
Second pass:

temp = 2
sum = 2 

|(2 - 2) - 2| = 2