Vyxal, 9 bytes

k-↔vÞż'∑¬

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Explained

k-↔vÞż'∑¬­⁡​‎‎⁡⁠⁡‏⁠‎⁡⁠⁢‏⁠‎⁡⁠⁣‏‏​⁡⁠⁡‌⁢​‎‎⁡⁠⁤‏⁠‎⁡⁠⁢⁡‏⁠‎⁡⁠⁢⁢‏‏​⁡⁠⁡‌⁣​‎‎⁡⁠⁢⁣‏‏​⁡⁠⁡‌⁤​‎‎⁡⁠⁢⁤‏‏​⁡⁠⁡‌⁢⁡​‎‎⁡⁠⁣⁡‏‏​⁡⁠⁡‌­
k-↔        # ‎⁡Get all combinations with replacement of the list [-1, 1] with length input
   vÞż     # ‎⁢multiply each item in each list by its one-indexed position
      '    # ‎⁣Filter to keep only lists where:
       ∑   # ‎⁤  The sum
        ¬  # ‎⁢⁡  logically negated is truthy (ie 0)
💎

Created with the help of Luminespire.

R, 82 69 bytes

s=t(expand.grid(rep(list(c(-1,1)),n<-scan())))*1:n;t(s[,!colSums(s)])

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Edit: -13 bytes by using built-in expand.grid function to automatically create matrix of all combinations of +1 and -1.

Selects rows of matrix of 1:n in all combinations of positive & negative which sum to zero.

Commented version:

s=t(                    # transpose matrix of:
mapply(rep,             # apply the function 'rep' 
    list(c(-1,1)),      # to the values (-1, 1)
    each=2^(0:(n-1)),   # repeating each by an increasing power-of-2
    length=2^n)         # and making the output length always 2^n
)                       # so: this generates a matrix of all the 
                        # combinations of -1 and 1
                        
t(                      # now transpose the matrix s
s[,                     # taking only the columns with a 
   !colSums(s*1:n)]     # zero sum of multiplying the values
)                       # by the sequence 1:n

Previous (but sadly longer much longer) recursive strategy: 118 bytes

f=function(n,l=NULL,t=0)`if`(n>1,c(f(n-1,c(0,l),t-n),f(n-1,c(1,l),t+n)),`if`(t==n,list(c(0,l)),if(t==-n)list(c(1,l))))

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Outputs 1 for + and 0 for - (or the other way around).

Commented version:

f=function(n,l=NULL,t=0){               # n = counts down to 1 recursively
                                        # l = list of signs so far
                                        # t = total so far
    if(n==1){                           # if n is 1...
        if(t-n==0){                     # ...then if t-n makes zero...
            return(list(c("-",l))) }    # add a '-' to the list of signs & return it
        else if(t+n==0){                # ...or, if t+n makes zero...
            return(list(c("+",l))) }    # add a '+' to the list of signs & return it
        else {                          # ...otherwise we can't make zero...
            return(NULL) }              # so return NULL
    } else {                            # if n is greater than 1
        return(c(                       # return the result of calling this function
            f(n-1,c("-",l),t-n),        # using n-1, after applying either a '-'
            f(n-1,c("+",l),t+n)         # or a '+' and updating the total so far
        ))
    }
}

T-SQL, 152 bytes

Input is a varchar(max)

Output format is +1+2-3

WITH C as(SELECT @*1d,@+null o,0f
UNION ALL
SELECT~-d,concat(char(44+s),d,o),f+s*d
FROM C,(values(-1),(1))x(s)WHERE d>0)SELECT
o FROM C WHERE f=0and d=0

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C (gcc), 101 100 83 bytes

Represents addition as 0 and subtraction as 1, but that ultimately that does not matter since for any valid sequence, the complement of the sequence will also result in zero. Returns the permutations by appending them as 32-bit ints to a caller-provided buffer.

i,j,s;f(n,p)int*p;{for(i=1<<n;--i;s||(*p++=i))for(j=s=0;j<n;)s+=i>>j++&1?j:-j;n=p;}

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Explanation

i,j,s; // iterators i and j, sum s.
f(n,p)int*p; // declare function
{
  for ( i=1<<n; --i; // iterate over all permutations of n bits (except all-0)
      s||(*p++=i)) // if the permutation summed to 0, append it to the buffer 
   for(j=s=0;j<n;)
     s+= i>>j++&1 // Determine the value of the jth bit
       ?j:-j; // 1=>+(j+1), 0=>-(j+1)
  n=p; // return the pointer to the end of the appended values
}

Not the best but I tried :)

C (Linux GCC)

Entire program: 282 characters

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
int n,x,i,s;int main(int c,char**z){n=atoi(z[1]);int a[n];for(x=0;x<n;x++)a[x]=x+1;for(i=0;i<pow(2, n)-1;i++){s=0;for(x=0;x<n;x++){printf("%c%d",i>>x&0x01?'-':'+',a[x]);s+=(i>>x&0x01?-1:1)*a[x];}printf(s!=0?"\r":"=%d \n",s);}}

Just as function: 196 characters

void z(int n){int a[n],x,i,s;for(x=0;x<n;x++)a[x]=x+1;for(i=0;i<pow(2,n)-1;i++){s=0;for(x=0;x<n;x++){printf("%c%d",i>>x&0x01?'-':'+',a[x]);s+=(i>>x&0x01?-1:1)*a[x];}printf(s!=0?"\r":"=%d \n",s);}}

Results

• n = 4:

+1-2-3+4=0 
-1+2+3-4=0

• n = 7:

+1-2-3-4-5+6+7=0 
-1+2-3-4+5-6+7=0 
-1-2+3+4-5-6+7=0 
+1+2-3+4-5-6+7=0 
-1-2+3-4+5+6-7=0 
+1+2-3-4+5+6-7=0 
+1-2+3+4-5+6-7=0 
-1+2+3+4+5-6-7=0

Abstract Printing & Just Function: 172 characters

You could shrink the print statements down much more by simply outputting only the +/-'s which are represented by 1/-1 and abstractly representing the entire sum thing, but I still don't think it would beat the other C answers anyway.

That version is:

void z(int n){int a[n],x,i,j,s;for(x=0;x<n;x++)a[x]=x+1;for(i=0;i<pow(2,n)-1;i++){s=0;for(x=0;x<n;x++){j=i>>x&0x01?-1:1;printf("%d ",j);s+=j*a[x];}printf(s!=0?"\r":"\n");}}

Results

• n = 4:

1 -1 -1 1 
-1 1 1 -1 

• n = 7:

1 -1 -1 -1 -1 1 1 
-1 1 -1 -1 1 -1 1  
-1 -1 1 1 -1 -1 1  
1 1 -1 1 -1 -1 1 
-1 -1 1 -1 1 1 -1 1 
1 1 -1 -1 1 1 -1 
1 -1 1 1 -1 1 -1 1 
-1 1 1 1 1 -1 -1 -1

JavaScript (V8),  69 61  58 bytes

Saved 8 bytes thanks to @Neil
Saved 3 bytes thanks to @l4m2

Prints all solutions.

f=(n,o='')=>n?f(n-1,o+'+'+n)&f(n-1,o+-n):eval(o)||print(o)

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C (gcc), 171 161 bytes

• Saved ten bytes thanks to ceilingcat.

k,s;f(S,n,j)int*S;{if(s=j--)S[j]=~0,f(S,n,j),S[j]=1,f(S,n,j);else{for(k=n;k;)s+=k--*S[k];if(!s)for(puts("");k<n;)printf("%d",S[k++]+1);}}F(n){int S[n];f(S,n,n);}

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Perl, 37 36 bytes

perl -E 'map eval||say,glob join"{+,-}",0..<>' <<< 7

J, 35 30 bytes

-5 bytes thanks to FrownyFrog!

>:@i.(]#~0=1#.*"1)_1^2#:@i.@^]

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Original:

J, 35 bytes

[:(#~0=+/"1)>:@i.*"1(_1^[:#:@i.2^])

How it works

I multiply the list 1..n with all possible lists of coefficients 1 / -1 and find the ones that add up to zero.

                    (             ) - the list of coefficients
                             i.     - list 0 to 
                               2^]  - 2 to the power of the input
                     _1^[:          - -1 to the power of 
                          #:@       - each binary digit of each number in 0..n-1 to 
                 *"1                - each row multiplied by
            >:@i.                   - list 1..n
  (#~      )                        - copy those rows
     0=+/"1                         - that add up to 0
[:                                  - compose   

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As an alternative I tried an explicit verb, using the approach of cartesian product of +/-:

J, 37 bytes

3 :'(#~0=+/"1)(-y)]\;{(<"1@,.-)1+i.y'

{(<"1@,.-) finds the cartesian products for example:

{(<"1@,.-) 1 2 3
┌───────┬────────┐
│1 2 3  │1 2 _3  │
├───────┼────────┤
│1 _2 3 │1 _2 _3 │
└───────┴────────┘

┌───────┬────────┐
│_1 2 3 │_1 2 _3 │
├───────┼────────┤
│_1 _2 3│_1 _2 _3│
└───────┴────────┘

Too bad that it boxes the result, so I spent some bytes to unbox the values

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Perl 6, 43 bytes

{grep *.sum==0,[X] (1..$_ X*1,-1).rotor(2)}

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Returns a sequence of lists

Expanded:

{  # bare block lambda with implicit parameter ｢$_｣

  grep              # only return the ones
    *.sum == 0,     # that sum to zero

    [X]             # reduce with cross meta operator

      (
          1 .. $_   # Range from 1 to the input

        X*          # cross multiplied by

          1, -1

      ).rotor(2)    # take 2 at a time (positive and negative)
}

1..$_ X* 1,-1 ⇒ (1, -1, 2, -2)
(…).rotor(2) ⇒ ((1, -1), (2, -2))
[X] … ⇒ ((1, 2), (1, -2), (-1, 2), (-1, -2))

Python 3 + numpy, 104 103 bytes

import itertools as I,numpy as P
lambda N:[r for r in I.product(*[[-1,1]]*N)if sum(P.arange(N)*r+r)==0]

Output is [-1, 1] corresponding to the sign.

Retina, 73 bytes

.+
*
_
=_$`
+0`=
-$%"+
(-(_)+|\+(_)+)+
$&=$#2=$#3=
G`(=.+)\1=
=.*

_+
$.&

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.+
*

Convert the input to unary.

_
=_$`

Convert the number to a list of =-prefixed numbers.

+0`=
-$%"+

Replace each = in turn with both - and +, duplicating the number of lines each time.

(-(_)+|\+(_)+)+
$&=$#2=$#3=

Separately count the number of _s after -s and +s. This sums the negative and positive numbers.

G`(=.+)\1=

Keep only those lines where the -s and +s cancel out.

=.*

Delete the counts.

_+
$.&

Convert to decimal.

Python 2, 62 bytes

f=lambda n,*l:f(n-1,n,*l)+f(n-1,-n,*l)if n else[l]*(sum(l)==0)

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Mr. Xcoder saved 4 bytes with a nifty use of starred arguments.

Clean, 79 bytes

import StdEnv
$n=[k\\k<-foldr(\i l=[[p:s]\\s<-l,p<-[~i,i]])[[]][1..n]|sum k==0]

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Pyth, 13 bytes

f!sT.nM*F_BMS

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APL (Dyalog), 38 bytes

{k/⍨0=+/¨k←((,o∘.,⊢)⍣(⍵-1)⊢o←¯1 1)×⊂⍳⍵}

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Python 2, 91 bytes

lambda x:[s for s in[[~j*[1,-1][i>>j&1]for j in range(x)]for i in range(2**x)]if sum(s)==0]

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Returns a list of satisfying lists (e.g., f(3)=[[-1,-2,3], [1,2,-3]])

Python 3, 105 bytes

lambda n:[k for k in product(*[(1,-1)]*n)if sum(-~n*s for n,s in enumerate(k))==0]
from itertools import*

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Husk, 10 bytes

fo¬ΣΠmSe_ḣ

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Explanation

Not too complicated.

fo¬ΣΠmSe_ḣ  Implicit input, say n=4
         ḣ  Range: [1,2,3,4]
     m      Map over the range:
      Se     pair element with
        _    its negation.
            Result: [[1,-1],[2,-2],[3,-3],[4,-4]]
    Π       Cartesian product: [[1,2,3,4],[1,2,3,-4],..,[-1,-2,-3,-4]]
f           Keep those
   Σ        whose sum
 o¬         is falsy (equals 0): [[-1,2,3,-4],[1,-2,-3,4]]

Haskell, 42 bytes

f n=[l|l<-mapM(\i->[i,-i])[1..n],0==sum l]

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Wolfram Language (Mathematica), 36 bytes

Pick[p={1,-1}~Tuples~#,p.Range@#,0]&

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Swift, 116 bytes

func f(n:Int){var r=[[Int]()]
for i in 1...n{r=r.flatMap{[$0+[i],$0+[-i]]}}
print(r.filter{$0.reduce(0){$0+$1}==0})}

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Explanation

func f(n:Int){
  var r=[[Int]()]                         // Initialize r with [[]]
                                          // (list with one empty list)
  for i in 1...n{                         // For i from 1 to n:
    r=r.flatMap{[$0+[i],$0+[-i]]}         //   Replace every list in r with the list
  }                                       //   prepended with i and prepended with -i
  print(r.filter{$0.reduce(0){$0+$1}==0}) // Print all lists in r that sums to 0
}

Jelly, 9 bytes

1,-ṗ×RSÐḟ

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Exp

1,-ṗ×RSÐḟ  Main link. Input = n. Assume n=2.
1,-        Literal list [1, -1].
   ṗ       Cartesian power n. Get [[1, 1], [1, -1], [-1, 1], [-1, -1]]
    ×R     Multiply (each list) by Range 1..n.
       Ðḟ  ḟilter out lists with truthy (nonzero)
      S      Sum.

Jelly, 9 bytes

Jonathan Allan's suggestion, output a list of signs.

1,-ṗæ.ÐḟR

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05AB1E, 11 bytes

®X‚¹ãʒ¹L*O_

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The output format for e.g. input 3 is:

[[-1, -1, 1], [1, 1, -1]]

That is, -1-2+3, 1+2-3.