This is a question with both a black and white answer -- and one that comes in a whole lot of shades of grey. Technically, when a ship misses a planned port of call due to mechanical difficulty, i.e. when the line is responsibility for the problem, it pays. What it has to pay is the port tax that was included in your cruise fare and typically cruise lines pony up another $50 - $100 per person. It's usually put on your onboard account rather than paid out in cash.
For any other reason -- weather, emergency illness requiring detour or evacuation, or any other factor beyond the cruise line's control -- it doesn't have to pay a cent. The disclaimer is part of the legalese that comes with your cruise ticket.
Carnival's situation -- the mutiny occurred on Carnival Conquest after the emergency medical evacuation of a crew member caused such a delay that the vessel had to skip a visit to Jamaica -- is unusual because the line actually did provide cruise travelers with a $25 onboard credit. Seems pretty skimpy but, compared with the nothing it was required to pay, it's at least something. In fact, Carnival actually has a written policy about missed port calls: "If safety, scheduling or other concerns prevent the ship from calling on an alternative port, Carnival shall promptly provide written notice of the elimination of the scheduled port to the passengers as well as announcing the change in the Carnival Capers and via the public address system. The written notice shall also offer a shipboard credit of $20.00 (U.S.D.) per person to be reflected on the sail and sign account."
It's important to remember that, while scheduled ports of call are certainly adhered to in most cases, you should be careful about scheduling once-in-a-lifetime events (such as weddings in port). If the ship doesn't make it -- you miss out on your special day (and probably won't get a refund of any deposits you made). Cruise lines won't reimburse you for that, either.
With regard to the cruise line, you have to separate the question into two discrete issues: What the cruise line will likely do, and what they are obligated to do. As for the latter, if you read your contract for passage (usually located in the last pages of your cruise documents) you may be shocked to learn how few obligations there are when it comes to weather or other factors out of the cruise line's control. However, when it comes to the former, most cruise lines will bend over backward to get you aboard with a minimum of delay, hassle or cost. After all, they would much rather have you onboard cursing the licentious slot machines than standing on the shore shaking your fist at the silhouette of a ship steaming toward the horizon.
How much they are willing to do and how much of the tab they'll pick up depends on a number of factors. For example, though weather-related travel delays are usually outside their area of obligation, unofficially, cruise lines are often more likely to go the extra mile for air/sea passengers than for those who booked their air independently.
Regardless, there are few circumstances where the cruise line will give you a refund because of itinerary changes or missed ports if the cause is weather-related.
Of course, that's what travel insurance is for. So, what will your insurance carrier do if a hurricane warning is issued for your destination? Access America will only compensate you if the destination is uninhabitable or unreachable. CSA, a little more liberal, will cover you if your destination is unreachable or there is a mandatory evacuation order. Travel Guard somewhat nebulously states that they will not cover you for hurricane warnings "unless [a] hurricane impacts travel or your destination." (Note: All these provisions were accurate at the time of this posting, but are subject to change.)
However, there are some aspects that no amount of compensation or accommodation can fix. Never, ever plan on cruising to a destination in the hurricane zone for a major event -- wedding, family reunion, etc. -- expecting the guests to arrive on their own and rendezvous with you there. Hardly a year goes by we don't hear from a Cruise Critic reader who arranged to cruise to Bermuda to be married, with all the family and guests booked to fly down there for the ceremony, only to find that a hurricane has forced the cruise to change the itinerary to Canada/New England!