In Ben-Hur, does the Sheik ever collect his 4000 talents?

Most certainly yes

There isn't a scene in the film that explicitly shows him collecting the debt because its not really important to the story, but that said, with what we know about the Sheikh and Messala as well as the scene where the bet is made, there is every reason to assume that he collected the money owed to him.

1. The Sheikh has a desire to beat Messala and humiliate him.
2. The Sheikh is a worldly man, with many wives, riches, servants, etc.
3. When the bet is made, Messala leaves a mark on a tablet with his ring, essentially the same as signing a contract.
4. Even though Messala dies, his estate would be obligated to settle the debt.

How much money was a talent?

Here is a discussion of the terms of the treaty that ended the First Punic War:

The next order of business was money. Rome wasn’t going to let Carthage walk away without handing over a large amount of talent. Carthage was to pay the Romans 3,200 talents over the next ten years. They also had to pay an indemnity of 1,000 talents immediately. Rome didn’t expect the Carthaginians to be able to pay the talents quickly. However, they paid it off much quicker than anticipated due to their new trade networks.

https://sites.psu.edu/cams101groupi2014/rome-responsibility-for-war/reparations-for-the-first-punic-war/1

So a peace treaty required one large country to pay another large country one thousand talents immediately, and then pay three thousand two hundred talents over a period of ten years, which would amount to a rate of 320 talents per year or 26.666 talents per month or about 0.876 talents per day.

So if the Romans thought that a thousand talents was a vast amount of money suitable as reparations after a war in which hundreds of thousands of Romans died, each talent must have been a very large sum of money.

So was there a lot of inflation between 241 BC and about AD 33, the approximate date of Ben-Hur?

Did individual aristocratic Romans tend to get a lot richer in those 275 years, and if so, did they get enough richer? Was Messala a member of the senatorial class or the equestrian class? How rich was Messala supposed to be compared to other Romans serving as tribunes?

It is my impression that neither the Sheikh nor Messala would have been able to pay off a bet of 1,000 talents or 4,000 talents.

But an ancient historian would be better qualified to answer that question.

A "Talent" is a unit of weight, not currency - Roman currency was called "Denarius".

From Wikipedia:

The talent was a unit of weight that was introduced in Mesopotamia at the end of the 4th millennium BC, and was normalized at the end of the 3rd millennium during the Akkadian-Sumer phase. It was divided into 60 minas, each of which was subdivided into 60 shekels. The use of 60 illustrates the attachment of the early Mesopotamians to their useful sexagesimal arithmetic. These weights were used subsequently by the Babylonians, Sumerians, and Phoenicians, and later by the Hebrews.[...]A Roman weight talent in ancient times is equivalent to 100 librae; a libra is exactly three quarters of an Attic weight mina, so a Roman talent is 1.33 Attic talents and hence approximately 32.3 kg (71 lb 3 oz).

So the bet sounds a bit like "I bet 1000 kg". Kilograms/Talents of what? If we indeed talking about Roman Talents of Gold/Silver then it is really doubtful that Messal could pay, if it was Talents of, let's say grain - quite probably.

But if we assume that indeed it was a reasonable value, then indeed Messala or whoever was taking from him would have to pay up.