Sometimes. Occasionally it can work its way out, down through the cervix. That’s called expulsion. Can be annoying, and means you don’t have birth control any more, but it’s not usually dangerous.
In rare cases, the IUD can work its way up, and into your abdomen. That’s called a perforation. That really doesn’t happen often, but if it does you would need a surgery to remove it. That’s why women are told to check for the strings: if you can feel the strings, everything’s probably okay.