A possible root cause of a tape addiction is pica, a the condition of cats eating inedible objects. Cats can develop pica if they are weaned to early.
In a word, yes. The problem is the possibility of intestinal blockage. Cats often throw up things they can’t digest, but not always, and a case of intestinal blockage because your cat swallowed a piece of tape could be life-threatening and costly.
Your first step should be to talk to a vet to make sure your cat isn’t showing signs of underlying cat health problems, but then you can focus on keeping your cat away from tape.
Try your best to keep tape or other adhesives out of reach of your cat – for example, don’t let them in the room when you are wrapping gifts! Try using other options, like twisty ties, whenever possible.