The main problem with your code is that you create a new PhotoImage for each pixel! Instead, create the PhotoImage once and just add the pixels in the double-for-loop.

def box(event):
    yaxis(event)
    canvas.create_rectangle(x1, y1, x2, y2)

    picture = PhotoImage(width=(x2-x1), height=(y2-y1))
    for x in range(x1, x2):
        for y in range(y1, y2):
            r, g, b = photo.get(x, y)
            picture.put("#%02x%02x%02x" % (r, g, b), (x-x1, y-y1))
    picture.write('new_image.gif', format='gif')

Also, the line tuple(map(int, value.split(" "))) in your getRGB function is wrong, as value is already the tuple you want to create, not a string.¹⁾ As you can see, I just 'inlined' that part directly into the box function. Another problem was that you wrote the copied pixels to x and y, but you have to write them to x-x1 and y-y1 instead.

Update 1: ¹⁾ It seems like the return value of PhotoImage.get depends on the version of Python/Tkinter you are using. In some versions, it returns a tuple, like (41, 68, 151), and in others, a string, like u'41 68 151'.

Update 2: As pointed out by @Oblivion, you can in fact just use the from_coords parameter of PhotoImage.write to specify the region of the picture to be saved to file. With this, the box function can be simplified as

def box(event):
    yaxis(event)
    canvas.create_rectangle(x1, y1, x2, y2)
    photo.write('new_image.gif', format='gif', from_coords=[x1, y1, x2, y2])