If you will be removing an IPH cartridge from the printer but may want to use the cartridge later do not put the tape back on the cartridge as this would ruin the cartridge. Opened cartridges should be left in the printer or stored as shown in the document here: https://support.hp.com/us-en/document/c03007680
Other printers have separate ink supplies and "plumbing" to connect to the printhead. These printers are known as Individual Ink Cartridge (IIC) printers. Some IIC printers have removable printheads that can be replaced if needed, others have permanent printheads. Most IIC printers will not allow printing if one or more of the ink cartridges are depleted. This is because allowing the printhead to run dry could cause permanent damage to the printhead, since ink provides necessary cooling to the printhead components. While some early IIC printers allowed printing with one or more colors out, most do not. The IIC printers that do not allow printing with a color out will typically give a warning when the ink is getting low. An option may be given to print with black only (if one of the color inks are low) or color only (if black is low the black areas will be printed with color inks). Once a color or black is completely depleted printing will be blocked until the depleted cartridge is replaced.
Note: when replacing a cartridge on an IIC printer the printer will typically go through a priming cycle to remove air that may have entered the ink delivery system. This priming cycle will use some ink, which may cause other colors or black that are nearly depleted to become depleted. I would suggest checking the ink levels before replacing a cartridge and replace all the cartridges that are nearly depleted at the same time. This will avoid running multiple priming cycles that will use extra ink in all the colors and black.
Caution: Do not remove an IIC cartridge for more than thirty minutes as this wilI allow air into the printhead and/or dry out the "plumbing". I have seen cases where folks remove a cartridge to take it to the store to make sure they get the proper replacement. This will often cause issues and should be avoided.