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I want to generate a list of random integers with only three values 1,0,-1. I know it can be done through RandomInteger[-1, 1, n]. How can I impose constraint on this list so that every integers in this list add up to 0?

(I have tried some approaches in Random numbers that sum up to specific value, but they did not work)

A combination of IntegerPartitions, RandomChoiceand RandomSample:

n = 30;
RandomSample @ RandomChoice @ IntegerPartitions[0, n, -1, 0, 1]

-1, 1, 1, 1, 1, -1, -1, 0, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1, -1, 0,
1, -1, -1, 1, -1, -1, 1, 1, -1, 1

Total @ %

0

You can also do

RandomSample @
 PadRight[Flatten @ ConstantArray[1, -1, RandomChoice[Range[0, Floor[n/2]]]], n]

-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, -1, -1, -1,
0, 0, -1, 0, 0, 0, 0, 1, 0

Total @ %

0

For large n, the second approach is much faster than the first.

You need more specificity. @kglr and @Alan give two possibilities (equal probabilities of partitions that sum to zero and (generally) equal numbers of -1's, 0's, and +1's, respectively).

If you want to consider a model where for any particular sample comes from a multinomial distribution with probabilities $p_-1$, $p_0$, and $p_1$ (subject to $p_-1+p_0+p_1=1$ and $0<p_-1<1$ and $0<p_0<1$ and $0<p_1<1$) and under the restriction that the sum of $n$ samples is zero can be found in the following manner:

(* Suppose we have the following: *)
n = 100;
p0 = 1/2;

(* Generate a number of zeros such that n - n0 is even and n0 is not negative *)
SeedRandom[12345];
n0 = -1;
While[n0 < 0 || OddQ[n - n0], n0 = RandomVariate[BinomialDistribution[n, p0]]];

(* The number of -1's and 1's much be each equal to (n-n0)/2 *)
x = If[n0 == 0, RandomSample[Join[ConstantArray[-1, n/2], ConstantArray[1, n/2]]],
 If[n0 == n, ConstantArray[0, n],
 RandomSample[Join[ConstantArray[-1, (n - n0)/2], ConstantArray[0, nZero], 
 ConstantArray[1, (n - n0)/2]]]]];
x
(* 0,1,0,-1,1,0,0,-1,1,0,1,-1,1,0,0,-1,0,0,1,0,1,-1,0,-1,0,-1,0,0,0,
 1,1,-1,0,0,1,0,0,-1,1,0,1,1,1,-1,-1,0,1,1,0,1,1,1,0,0,-1,1,0,0,0,0,-1,
 -1,-1,1,0,-1,0,0,0,1,1,-1,0,-1,1,1,-1,0,-1,0,0,0,0,-1,0,-1,0,0,0,0,-1,
 0,-1,-1 *)

Total[x]
(* 0 *)

Notice that because the sum is required to be zero, any values of $p_-1$ and $p_1$ result in the same restricted random distribution once $p_0$ is set.

If you want to sample uniformly from all possible tuples that sum to 0, you can do the following:

zero[n_] := With[
 
 ones = RandomChoice[
 Table[Multinomial[i,i,n-2i], i,0,Floor[n/2]] -> Range[0,Floor[n/2]]
 ]
 ,

 RandomSample @ PadRight[
 Join[ConstantArray[1,ones],ConstantArray[-1,ones]],
 n
 ]
]

For example, here's a tally of the random 4-tuples summing to 0:

SeedRandom[1];
Tally @ Table[zero[4], 10^5]

1, 1, -1, -1, 5215, -1, 1, 0, 0, 5353, -1, -1, 1, 1,
5381, 1, -1, 0, 0, 5167, 1, -1, -1, 1, 5169, 0, 1, 0, -1,
5189, 0, -1, 1, 0, 5311, -1, 1, -1, 1, 5263, 0, -1, 0, 1,
5376, 0, 0, 1, -1, 5268, 1, 0, -1, 0, 5303, 0, 0, 0, 0,
5218, 1, 0, 0, -1, 5220, 1, -1, 1, -1, 5095, 0, 0, -1, 1,
5245, -1, 0, 0, 1, 5313, -1, 1, 1, -1, 5253, 0, 1, -1, 0,
5264, -1, 0, 1, 0, 5397

Looks pretty close to uniform sampling.

As a comparison, note the distribution using the accepted answer:

nonuniform[n_] := RandomSample @ PadRight[
 Flatten@ConstantArray[1,-1,RandomChoice[Range[0,Floor[n/2]]]],
 n
]

Tally @ Table[nonuniform[4], 10^5]

-1, 0, 1, 0, 2739, 0, 0, 0, 0, 33695, 0, 1, 0, -1,
2771, -1, -1, 1, 1, 5682, 0, 0, 1, -1, 2807, 1, -1, 0, 0,
2697, 0, 0, -1, 1, 2790, -1, 0, 0, 1, 2765, -1, 1, 0, 0,
2738, 0, -1, 0, 1, 2654, -1, 1, -1, 1, 5482, 1, -1, -1, 1,
5555, 0, -1, 1, 0, 2768, 1, 0, -1, 0, 2721, -1, 1, 1, -1,
5519, 1, 1, -1, -1, 5512, 0, 1, -1, 0, 2727, 1, 0, 0, -1,
2710, 1, -1, 1, -1, 5668

Not very uniform.

Given the lack of details in your specification, I suspect the following very simple approach will be adequate to your needs:

zeroSum[n_] := With[
 n3 = Floor[n/3]
 ,
 RandomSample@Catenate@
 ConstantArray[-1, n3],
 ConstantArray[1, n3],
 ConstantArray[0, n - 2*n3]
 ]