Many things inside of a tattoo and piercing shop are considered biological hazards. These include things that have come into contact with open skin, or have touched a bodily fluid. This includes, but is not limited to: used needles, tubes, piercing tools, clamps and tapers. All of these things should be placed inside of a sharps containers, biohazard labeled bins and trash receptacles after use.
As for blood soaked trash, in the US, most states have different rules as to whether the material must be treated as biohazardous or simply contaminated waste. Any local reputable piercing/tattoo studio should be able to answer your questions regarding material disposal.