For how long after finishing a course of antibiotics does the medication continue to work/have an effect against infection?

Not very long: Antibiotics such as antibacterials or antivirals don't last that long in the body. Often, these medications are chemicals that patients take orally, to interfere with the life and reproduction of bacteria or viruses. Our bodies do a good job of breaking down and removing chemicals that we swallow, such as antibiotics. Most effects fade after a couple of days. Some (like Zithromax) can last a week.