Bash, 46 bytes

ImageMagick scales image to one pixel which contains average of the colors in the image then outputs it as text.

convert $1 -scale 1x1\! txt:-|egrep -o '#\w+'

MATLAB - 68 bytes

The image is read in with imread combined with uigetfile to open up a GUI to choose the image you want to load in. The assumption with this code is that all images are RGB, and to calculate the average colour, we sum over each channel individually then divide by as many elements as there are in one channel, which is the total number of pixels in the RGB image (numel(I)) divided by 3. Because the average can possibly generate floating point values, a call to fix is required to round the number down towards zero. sprintf combined with the hexadecimal formatting string (%x) is used to print out each integer value in the average into its hex equivalent. However, the 02 is there to ensure that an extra 0 is padded to the left should the average value for any channel be less than 16^*.

I=imread(uigetfile);
['#' sprintf('%02x',fix(sum(sum(I))*3/numel(I)))]

Sample Runs

The nice thing about imread is that you can read in images directly from URLs. As a reproducible example, let's assume you have downloaded the images on your computer and have run the above code... but for demonstration, I'll read the images directly from Code Golf:

First Image

>> I=imread('https://i.sstatic.net/dkShg.jpg');
>> ['#' sprintf('%02x',fix(sum(sum(I))*3/numel(I)))]

ans =

#53715f

Second Image

>> I=imread('https://i.sstatic.net/V5VAR.jpg');
>> ['#' sprintf('%02x',fix(sum(sum(I))*3/numel(I)))]

ans =

#8b7d41

^{*Note: This was a collaborative effort made by StackOverflow users on the MATLAB and Octave chat room.}

CJam, 27 bytes

'#[q~]5>3/_:.+\,f/"%02X"fe%

This read a PPM image from STDIN.

CJam has no built-in image processing, so this code expects an ASCII Portable PixMap (magic number P3) with full 24-bit palette (maximum value 255) and no comments.

Test run

$ cjam avg.cjam < dkShg.ppm 
#53715F

How it works

'#     e# Push that character.
[q~]   e# Evaluate the input and collect the results in an array.
5>     e# Discard the first first results (Pi, 3, width, height, range).
3/     e# Split into chunks of length 3 (RGB).
_:.+   e# Push a copy and add across the columns (RGB).
\,f/   e# Divide each sum by the length of the array (number of pixels).
"%02X" e# Push that format string (hexadecimal integer, zero-pad to two digits).
fe%    e# Format each integer, using the format string.

HTML5 + JavaScript (ES6), 335 bytes

This is not going to win but I had fun doing it anyways.

Uses the HTML5 Canvas API. Input is a URL of a CORS-enabled image.

f=(u,i=new Image)=>{i.crossOrigin='';i.src=u;i.onload=e=>{x=(c=document.createElement('canvas')).getContext('2d');a=w=c.width=i.width;a*=h=c.height=i.height;x.drawImage(i,0,0);for(d=x.getImageData(m=0,0,w,h).data,r=[0,0,0];m<d.length;m+=4){r[0]+=d[m];r[1]+=d[m+1];r[2]+=d[m+2]}console.log('#'+r.map(v=>(~~(v/a)).toString(16)).join``)}}

Demo

As it is ES6, it currently only works in Firefox and Edge.

f = (u,i = new Image) => {
  i.crossOrigin = '';
  i.src = u;
  i.onload = e => {
    x = (c = document.createElement('canvas')).getContext('2d');
    a = w = c.width = i.width;
    a *= h = c.height = i.height;
    x.drawImage(i, 0, 0);
    for (d = x.getImageData(m = 0, 0, w, h).data, r = [0, 0, 0]; m < d.length; m += 4) {
      r[0] += d[m]
      r[1] += d[m + 1];
      r[2] += d[m + 2];
    }
    console.log('#' + r.map(v => (~~(v/a)).toString(16)).join``)
  }
}

// Snippet stuff
console.log = x => document.body.innerHTML += x + '<br>';

f('http://crossorigin.me/https://i.sstatic.net/dkShg.jpg');

f('http://crossorigin.me/https://i.sstatic.net/V5VAR.jpg');

Python [3] + SciPy, 144 133 121

Loads pixel data, sums for each channel, divides by size*, formats. Values are rounded towards zero.

_{^{*size = width * height * channels, thus multiplied by 3}}

from scipy import misc,sum
i=misc.imread(input())
print('#'+(3*'{:2x}').format(*sum(i.astype(int),axis=(0,1))*3//i.size))

R, 90 bytes

rgb(matrix(apply(png::readPNG(scan(,"")),3,function(x)sum(x*255)%/%length(x)),nc=3),m=255)

Path to PNG file is read from STDIN. Package png needs to be installed.

Step-by-step:

#Reads path from stdin and feeds it to function readPNG from package png
p = png::readPNG(scan(,""))
#The result is a 3d matrix (1 layer for each color channel) filled with [0,1] values
#So next step, we compute the mean on each layer using apply(X,3,FUN)
#after moving the value to range [0,255] and keeping it as an integer.
a = apply(p,3,function(x)sum(x*255)%/%length(x))
#The result is then moved to a matrix of 3 columns:
m = matrix(a, nc=3)
#Which we feed to function rgb, while specifying that we're working on range [0,255]
rgb(m, m=255)

# Example:
rgb(matrix(apply(png::readPNG(scan(,"")),3,function(x)sum(x*255)%/%length(x)),nc=3),m=255)
# 1: ~/Desktop/dkShg.png
# 2: 
# Read 1 item
# [1] "#53715F"

C, 259 Bytes

Takes a PPM file with no comments.

double*f,r,g,b;x,y,z,i;char*s="%d %d %d";main(a,_){(a-2)?(feof(f)?0:(fscanf(f,s,&x,&y,&z),r+=(x-r)/i,g+=(y-g)/i,b+=(z-b)/i++,main(0,0))):(f=fopen(((char**)_)[1],"r"),fscanf(f,"%*s%*d%*d%*d"),r=g=b=0.,i=1,main(0,0),printf(s,(int)r,(int)g,(int)b),fclose(f));}

Process

Initial code:

#include <stdio.h>
#include <stdlib.h>

int main(int argc, char *argv[])
{
    FILE *f = fopen(argv[1],"r");
    int w,h,d,x,y,z,i;
    double r,g,b;
    fscanf(f,"%*s %d %d %d",&w,&h,&d);//get width, height, depth, ignore P6
    r = g = b = 0.0; //zero r, g, and b totals
    for (i=1; i<=w*h; ++i) {
        fscanf(f,"%d %d %d",&x,&y,&z);//get next pixel
        r+=(x-r)/i;//update averages
        g+=(y-g)/i;
        b+=(z-b)/i;
    }
    printf("%d %d %d",(int)r,(int)g,(int)b);//print result
    fclose(f);
    return 0;
}

Trim variables and remove loop:

double r,g,b;
FILE *f;
int i;
int main(int argc, char *argv[])
{
    if (argc==2) { // {./me} {file.ppm}
        f = fopen(argv[1],"r");
        fscanf(f,"%*s%*d%*d%*d");//drop info
        r = g = b = 0.0;
        i = 1;
        main(0,0);//begin load loop
        printf("%d %d %d",(int)r,(int)g,(int)b);
        fclose(f)
    } else {
        if (feof(f)) return 0;
        fscanf(f,"%d%d%d",&x,&y,&z);
        r+=(x-r)/i;
        g+=(y-g)/i;
        b+=(z-b)/i;
        i++;
        main(0,0);
    }
    return 0;
}

From there I combined the various statements into a single return statement. Removed it and any other extraneous type info, renamed variables, and cut the whitespace.

Cobra - 371

@ref 'System.Numerics'
use System.Drawing
use System.Numerics
class P
    def main
        i,d=Bitmap(Console.readLine?''),BigInteger
        r,g,b,c=d(),d(),d(),d()
        for x in i.width,for y in i.height,r,g,b,c=for n in 4 get BigInteger.add([r,g,b,c][n],d([(p=i.getPixel(x,y)).r,p.g,p.b,1][n]))
        print'#'+(for k in[r,g,b]get Convert.toString(BigInteger.divide(k,c)to int,16)).join('')

Java, 449 447 446 430 426 bytes

import java.awt.*;interface A{static void main(String[]a)throws Exception{java.awt.image.BufferedImage i=javax.imageio.ImageIO.read(new java.io.File(new java.util.Scanner(System.in).nextLine()));int r=0,g=0,b=0,k=0,x,y;for(x=0;x<i.getWidth();x++)for(y=0;y<i.getHeight();k++){Color c=new Color(i.getRGB(x,y++));r+=c.getRed();g+=c.getGreen();b+=c.getBlue();}System.out.printf("#%06X",0xFFFFFF&new Color(r/k,g/k,b/k).getRGB());}}

Thanks to this answer over on stack overflow for the String.format trick.