In every instance I've ever heard of, they are as good as any airline when it comes to IRROPS.

If it's a mechanical issue or misconnect, they rebook you (space available, of course, but that's the same as any airline) on the next available train to your destination. If necessary, they'll put you up in a hotel if that requires an overnight. In some cases involving misconnects, I've seen them run a bus to "catch up" to the connecting train or even (in admittedly rare circumstances) pay for a taxi to get passengers to their final destinations. I think even once (but just once) I heard about Amtrak booking a delayed passenger on an airline to their final destination, but I would not ever count on that!

While Amtrak trains are affected by weather much more rarely than airlines, in the rare cases where weather causes delays, Amtrak will still rebook you on the next available train, but (just like airlines), they probably won't cover hotel costs (if applicable).

A couple notes about rebooking: just like airlines, they will rebook you space available. If you're in a sleeper compartment or business/first-class seat and they are booked solid, Amtrak will offer you the option to take a coach seat and receive a refund of the difference. On shorter-distance trains (like Northeast Corridor services), Amtrak can do something an airline can't: stuff more people on a train than there are seats. While it's not the most comfortable option, most people would rather get to where they're going even if they have to stand for an hour or two or sit on a cafe car stool instead of a seat than be stuck in a train station (or airport). In these cases, while it's not an official policy, conductors will often not "lift" the ticket of those passengers who are not able to find a seat. This means you are then free to seek a refund (90% of the ticket's value) or exchange your ticket for another one (100% of the ticket's value), effectively giving you a free ride during the IRROPS.

The one downside of Amtrak: Amtrak does not have (as far as I know) an actual standby procedure (like airlines), so I think you have to take the first rebooking option available that a ticket agent presents you with. The standby procedure consists of walking out to each train and seeing if the conductor will take your ticket for his/her train...

(Without a live, electronic ticketing system, Amtrak has no feasible way to track no-shows, so a standby list to fill empty seats is really kind of impossible. Such a system is reportedly in the works, however, so we may see this develop over the next few years.)