There are two levels of team captain powers, official powers / rights and powers that his/her national federation confers. Examples of these would be that a team player may ask his team captain if he may accept a draw offer. The captain's advice is non-binding. This is an official power. Normally players are not allowed to discuss their games with anybody and asking advice about a draw offer would count as such.
Note that the captain is not allowed to remind the player to press the clock and strictly speaking he shouldn't even tell the player his opponent has moved if the player has gone to the toilet or for a coffee.
Agreeing a draw on somebody else's board is a power that the national federation confers (or not) on the captain and the player involved acquiesces to this by accepting a place in the team. There have been cases where in the last round a team just needs a draw (2-2) to win or secure a particular position and even though they are playing a weaker team they come out, play a few moves and then everyone agrees a draw.
Deciding who plays and who rests is usually a question for the manager and captain although obviously players can also have input. "Captain, I'm absolutely knackered. I need a day off. Give me a rest, please" or "I just want to play every game. Rest somebody else." Ultimately these are management decisions and the captain and the manager will make the final decision.
FWIW a few years ago my team captain (at a level at least a dozen levels below Olympiad, I hasten to add) used to tell me I had to refuse draw offers if we were losing, regardless of the position on my board. Needless to say I didn't pay much attention!
Here are the official FIDE rules governing the role of the captain:
- Team Captain’s Role in Team Tournaments
(a) The role of a team captain is basically an administrative one during play. Depending on the regulations of the specific competition, the captain shall be required to deliver at a specific time a written list naming the players in his team participating in each round, to communicate to his players their pairing, to sign the protocol indicating the results in the match at the end of the play, etc.
(b) Whenever the team captain speaks to one of his players, he should do so only through or in the presence of an arbiter, using a language the arbiter can understand.
(c) A captain is entitled to advise the players of his team to make or accept an offer of a draw or to resign a game, unless the regulations of the event stipulate otherwise. He must confine himself only to brief information, based solely on the circumstances pertaining to the match. He may say to a player, “offer a draw”, “accept the draw”, or “resign the game”. For example, if asked by a player whether he should accept an offer of a draw, the captain should answer “yes”, “no”, or delegate the decision to the player himself. He shall give no information to a player concerning the position on the chess board and/or the clock times, nor consult any other person and/or computer as to the state of the game. The captain shall refrain from any intervention during play.
(d) Players are subject to the same prohibitions. Even though in a team competition there is a certain team loyalty, which goes beyond a player’s individual game, a game of chess is basically a contest between two players. Therefore a player must have the final say over the conduct of his own game. Although the advice of the captain should weigh heavily with the player, the player is not absolutely compelled to accept that advice. Likewise, the captain cannot act on behalf of a player and his game without the knowledge and consent of the player.
(e) A team captain should encourage his team always to follow both the letter and the spirit of Article 12 of the FIDE Laws of Chess concerning the conduct of the players. Team championships, in particular, should be conducted in the spirit of the highest sportsmanship.