Since August 15, the Feast of the Assumption, is a Holy Day, the Vatican Museums will be closed that day. However, the archaeological sites in Rome will be open. So if you will be in Rome on the 15th that means you can visit the Colosseum, the forum, etc. since Rome is part of Italy and does not necessarily observe the religious holiday.
The Vatican, obviously, is its own country and therefore has its own rules. The Vatican Post Office will also be closed, but both will be open again on the 16th. Major Basilicas (including Saint Peter’s) and Churches will be open and probably crowded due to the feast day.
And one other note: Pope Francis will be at the Papal Summer residence at Castel Gandolfo that day. Whether he will make a public appearance or not is unknown, but our guess is he probably will.