Depends on the cause. There are different parts of the eye that many people call "tear ducts" but may not be the actual tear duct. The tear duct, which drains your tears, can get blocked for reasons that are generally not helped by eye drops. A clogged tear duct usually has to be probed or even have surgery to unclog it. If you are referring to the pores that line the lids that can get clogged, abx can help.
Yes but. If the cause is from debri from bacterial conjubctivitis this can help clear without surgery. If it is viral or severe allergic reaction or certain types of bacteria it can actually close off the opening or even further down. This can be like getting a leaf stuck at the top of a 2 story gutter.