How to print to a hp officejet j5780 printer without the color cartridge

Many inkjet printers actually even use (and thus: require) colour when printing in greyscale. If the colour cartridge is empty, such printers might refuse to print in greyscale, and thus will not print images. (And those might even not print black text when selecting greyscale.)

When you don't need colour, get a black and white printer instead.

Odd things have been reported. Like for some brands printing in black and white using best quality avoids using the colour cartridge, while selecting some economy mode would actually use colour:

Is there a driver setting [..] to print photos using the black cartridges for the printer?
[..]
Until the iP4500, it was enough to check "Grayscale" and select "plain paper" and "high quality".

And, things might be much more complicated, like for the Canon MP760 and the iP4000, which use a large BCI-3e pigment black cartridge, and smaller BCI-6 dye inks including yet another black (and yellow, cyan and magenta):

The truth is that the black pigment ink is [almost] always used on plain paper--never on photo paper and the dye black ink is always used on photo paper--[almost] never on plain paper.
[..]
The exception is when doing borderless printing which is not recommended on plain paper, but if you do it, the printer will use the dye black ink instead of the pigment black ink.

Pretty simple, but there is a complication involving duplex printing. When doing duplex printing, the black pigment ink is cut to about one half of the intensity that it would normally print. Then, to make up for this decrease, the magenta and cyan inks are printed in the same area as the pigment black. The yellow and dye black are never added to the pigment black to make up the difference. NOTE: Recent tests have shown that this last statement is only true for 100% black. In fact, yellow, along with magenta and cyan, is blended for some shades of grey. Also, it doesn't matter whether the duplex printing is done automatically or manually, or whether you are doing booklet printing. And, in all cases, we are talking about duplex printing on plain paper.

(As a side note: some laser printers are known to count colour usage even when printing in black and white.)

You will not get anything on the paper with a basic CMYK inkjet or laser printer. The CMYK color mixing is subtractive, meaning that it requires the base that is being colored to have all colors (i.e., White) So that it can create color variation through subtraction:

White - Cyan - Yellow = Green
White - Yellow - Magenta = Red
White - Cyan - Magenta = Blue

White is represented as 0 cyan, 0 yellow, 0 magenta, and 0 black - effectively, 0 ink for a printer that simply has those four cartridges. This works great when you have white media, as "printing no ink" simply leaves the white exposed, but as you can imagine, this doesn't work for non-white media.

If you don't have a base color to subtract from (i.e., Black), then it doesn't matter what you subtract from it, you still have the color Black.

As others are pointing out, there are special printers which can operate in the CMYW color space, or otherwise have a white ink or toner. These can be used to print light colors on top of dark or otherwise non-white media.

You might also find my answer to a different question about color spaces helpful or informative.