"Add one" to every color in an image

Mathematica, 78 bytes

Image@Apply[16#+#2&,Mod[IntegerDigits[#~ImageData~"Byte",16,2]+1,16]/255,{3}]&

Takes and returns an image object (to create an image object, just paste the image into Mathematica).

Result for the test case:

Taking input and returning output as a 3D array of integer channel values, this reduces to 51 bytes:

Apply[16#+#2&,Mod[IntegerDigits[#,16,2]+1,16],{3}]&

But those meta posts don't have an overwhelming amount of support yet, so I'm going with the 78-byte version for now.

Pyke, 17 13 bytes

.Fh16j%ijcjs 8{

Try it here!

.F            - for i in deep_for(input):
  h16%        -    (i+1)%16
          +   -   ^+V
      i16+    -    (i+16)
           8{ -  unset_bit(8, ^)

Takes input as a 3d integer array of pixels and outputs in the same format

Python, 226 bytes

Now, it's valid !

Use the Pillow library.

from PIL import Image
m=Image.open(input()).convert("RGB")
for y in range(m.size[1]):
 for x in range(m.size[0]):
    t=m.getpixel((x,y))
    h=t[0]+(t[1]<<8)+(t[2]<<16)+1118481
    m.putpixel((x,y),(h&255,h>>8&255,h>>16&255))
m.show()

Output:

Thanks to @TuukkaX for saving 9 bytes !
Thanks to @mbomb007 for saving 18 bytes !