Does anyone know what type of tape will be kept on the longest without
leaving its glue? [Even if I can't find something that will last 3
months without depositing its residue, learning about something that
will last longer than duck tape will be useful].

The blue masking tape can be left on for at least a month, I'd say,
without leaving a residue. But it won't stick to a carpet. It might
work to stick it to the baseboard.