What to look for when it keeps saying ... Error .... Remove and check color Cartridge?

If you want remove errors, you can simply check the Input Sign on the Cartridge to fix it insie the Photosmart Printer.

Then you can use your Photosmart 2610 printer.

You could be getting this problem if using refilled inkjet cartridge. Try replacing cartridge and see if message clears.

You can try buffing the contacts gently but this happens quite often with generic cartriges sometimes they just don't work. Contact the vendor and explain the issue, I reckon you'll find that they're fairly used to this problem and will request a return for a refund.

try cleaning cartridge carrier with a wet towel 

Have you tried replacing the cartridges all together? That worked for me. I use HP 96 and 97 cartridges.

Take out cartridge and clean the contacts. On the printer and on the cartridge. Mine did that and there was a speck of ink where the cartridge meets the printer contact. Actually, it was a defective cartridge.

try cleaning the contacts opd the color cartridge 
with a wet towel 
if the problem is not solve return the cartridge in the shop you get it and ask them to replace it

does it print with the color cartridge

the problem is the little poorly designed metal thing that is holding up the ink cartridge get's stuck between the cartridge and the printed circuit and it brakes the circuit.must have been some **** drawering on a napkin that came up with that one.wasted a nice unit for that two cent piece of metal.