…maybe also Cornwall?
These are estates from which the duke derives some sweet revenue. These monies may have some official purpose, but that is not the point. My question is, are there any other situations where you personally get a bunch of cash just by virtue of being named Duke of Whatever (or some other title)?
It’s not cash, but the Bishop of Urgell also gets to be Co-Prince of Andorra.
Does a (co-)Prince of Andorra even get a salary? 
If so, I suppose that counts. But then we have to admit that, not sure about Andorra, but in lots of places being head of state comes with a salary. But is there some arrangement where you are not the head of state (king or whatever) but still derive that kind of income?
So AFAIK it’s not salary, it’s the income from those lands, in terms of rent etcetera. That’s how aristocratic holdings work, since feudal times (or at least early modern times, the defining feature of feudalism being that the peasants paid their lord in-kind not cash)
Right; I was especially interested in a (hypothetical) The Prince and the Pauper kind of situation where, in the story, “Also learn, that for a higher service… he is a peer of England, Earl of Kent, and shall have gold and lands meet for the dignity.” But those lands and their gold cannot come from thin air… was the king supposed to be enfeoffing him out of his own feudal holdings?
I mean that is the fundamental calculus a monarch has to work out*. They need to dole out lands and titles to their supporters but they have to come from somewhere, either from crown lands (reducing their income) or by taking it off some other aristo who is not in their good books (risking driving them into rebellion)
* - historically I don’t think king Charles is too concerned his marcher lords are going to rebel unless he gives them a nice chunk of countryside and a fancy new title 
Yes, a Dukedom (“dukeage”?) basically makes you the one who “owns” the ands associated with the title. Many may have frittered away much of their assets over the generations. (I recall something about some parts of the estates belonging to the title, not the person, so not available for disposal by the title-holder)
Also, more modern titles bestowed/bought do not come with the same level of perks as the hereditary ones. They are more a formality than a sinecure. (Modern peerrages IIRC are generally not hereditary also)
Yeah, it’s really just being a landlord on a very large scale.
Which is as good a time as any to point out that, technically, King Charles is a McDonald’s franchisee.
No. From your cite
the Crown Estate rents out space to seven McDonald’s restaurants, four Starbucks stores, and five Subway stores
Landlord, not franchisee
It’s worth pointing out that the Duchy of Lancaster estates like the Crown estates are “held in trust” and managed by trustees accountable to Parliament (the ancient title of Chancellor of the Duchy of Lancaster is still used for a senior cabinet member that the PM wants to handle some cross-departmental policy issue, though they might still have some formal duties associated with the Duchy).
The income from the Duchy is at the monarch’s disposal, but he can’t just give away the capital.
The same applies to the Crown Estates (the bulk of that income goes to the Treasury for general government purposes) and the Duchy of Cornwall (the Duke and his trustees are prohibited by law from doing anything in the management of the estates that reduces the capital, without the approval of Parliament).
Without going into all the intricacies of the modern (since 1960s) non-hereditary life peerage, I think it’s a fair summary to say that they are the UK’s equivalent to the various medals of merit and other public accolades existing in other countries. The hereditary peerages have not formally been abolished, but they’re only awarded within the royal family nowadays. Even if one is awarded today, it doesn’t come with any perks other than the right to use the title; there are no landholdings attached to a newly created peerage or anything like that. The life peerage, on the other hand, comes with a special perk: an automatic seat in the House of Lords (this is why they had to make David Cameron a life peer when Sunak wanted to make him a minister: You have to sit in either the Lords or the Commons to be a member of Cabinet).
I am vaguely remembering an anecdote about Bertrand Russell. His brother died, which made him an earl. When asked about this, he replied that the title might be useful getting tables at restaurants. Perhaps someone can dig up the exact quote? Point is, it did not sound like he got “lands and gold”.
Some do, some don’t. Depends on whether their ancestors frittered away the family gold.
Fun fact: Russell’s brother Frank was also the last peer to be convicted by the House of Lords sitting as a criminal court. The right of peers to be tried by the House of Lords was an ancient privilege that was only abolished in 1948.
Always remember to pass the Duchy ‘pon the left hand side.
The “getting a table” thing is a well-established trope, not just his idea. His earldom came because a 19th century ancestor had been Prime Minister, as was the custom.
The automatic link between a title and “lands and gold” was a mediaeval thing, which died out, I suppose as power moved away from the monarch to Parliament. Titles became rewards for past services in government, which was in the hands of people who already had lands and gold, rather than a delegation of royal power.
Trivia note: While King Charles’ mother was a Queen (not a King) she was Duke (not Duchess) of Lancaster. Traditionally most titles (but NOT the Monarchy) could only be inherited by a male, so she was called Duke to clarify that she was the title-holder, rather than just married to a Duke as “Duchess” would imply.
For the same reason, Elizabeth II was also Duke of Normandy and Lord of Mann.
Although we are drifting somewhat from the Duchy of Lancaster, most (non-royal) peers’ title has no connection with the lands they actually own. The Duke of Devonshire’s estate is Chatsworth in Derbyshire. The Earl of Derby’s powerbase is around Liverpool. The Marquis of Salisbury’s seat is Hatfield House in Hertfordshire, the family also have extensive lands around Cranborne in Dorset and spend part of their time there.