Which websites provide last-minute deals for trains in Europe?

I don't think such a thing exists, because unlike airline tickets or hotel rooms, train tickets aren't generally sold as a limited resource - for most European trains you can walk up to the station and buy a ticket 5 minutes before the train leaves, at full price. As these on-the-day tickets aren't usually tied to a particular train or seat, they also don't know exactly how many people will be on a given train, and this can often be more than the number of seats.

Last minute deals on train tickets are not generally allowed - for instance, such deals are prohibited by ORCATS (the inter-company ticket settlement system) rules in the UK. Discount tickets in the UK must be available from when reservations are first made available (normally 90 days before departure). This is why the prices rise, as the cheapest class of discount tickets steadily sells out.

Trains also don't usually require reservations, so a train ticket (at least certain types of tickets) just lets you jump on any train you like. For this reason, train companies, unlike airlines, don't know exactly how many spare seats they have on a train. Even for the few registration-required services (like Eurostar or the TGV), there are still lots of passengers with fully-flexible tickets who could switch their booking to that train at the last minute, so there's a risk in selling off the last seat cheaply: a full-price passenger wishing to switch to that train then can't get on - which would be an expensive mistake for the train company.

None of this applies to certain private rail companies that run their own trains separately from the main rail system. Mostly these are railtours (often these use vintage locomotives or carriages or both: some are really expensive, like the Orient Express, but others are more modestly priced) and some of them, especially the ones that aren't deluxe holidays, will offer last minute tickets at a discount - but many of these are small companies and don't have the sort of sophisticated ticketing systems that have the ability to do this sort of thing.