What will happen if I will not change our filter cartridge for a very long time?

You have this under 'food and drink' so I am assuming you mean the filter in your water container. Nothing will happen, your water will not be filtered any longer through charcoal that is not useable after a certain time.

If you mean an in-line filter, pretty much the same.
Suggested times to change are for a family of 4. Mine is supposed to be changed twice a year, but since I live alone, I change it every 2 years.
Mine is a string filter, not charcoal. The string is white so I can tell from the color and flecks of minerals and metals how clogged it is.