Can I fix damaged AC line insulation with tape?

Generally speaking, if you need to tape something HVAC you should use foil tape, not other tape. While it won't "harm" anything to use other tape, foil tape is made from metal so it forms a new solid connection.

You should not tape this pipe. The foam insulates the cold pipe so it doesn't leak its coldness into the warm air outside, wasting energy. It also prevents the cold pipe from condensing water from the air around it. The foam insulation is cheap and is very easy to install. Just buy some on line and snap it onto your pipe. You can use a little tape over the foam just to hold it on but not instead of the foam.

Can you run your AC without the foam? You can, just look at the part of the pipe inside the house. Without the foam insulation it may be condensing and dripping water so just make sure it isn't dripping somewhere you don't want.