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In this precise moment of this video, (found on this this article) Donald Trump says about Greenland:

It's hurting Denmark very badly because they are losing almost $700m a
year carrying it, so they carry it at a great loss

The fact that Greenland is subsidized by Denmark is on CIA world factbook (inside the "economy" tab):

Greenland’s economy depends on exports of shrimp and fish, and on a
substantial subsidy from the Danish Government. Fish account for over
90% of its exports, subjecting the economy to price fluctuations. The
subsidy from the Danish Government is budgeted to be about $535
million in 2017, more than 50% of government revenues, and 25% of GDP.

Is it fair to say that Denmark has to "carry" Greenland, or do the benefits/profits outweigh the costs?

Trump's claim isn't too far off from the subsidy Denmark pays to Greenland, but it's off by $150 million. Moreover, his characterization of the subsidization as a "loss" misrepresents Greenland's economic value, and drastically overstates the importance of the subsidy to Greenland.

To check the CIA Factbook number, I looked at an government publication of Greenland's statistics (published by Greenland's government). In the Økonomi ("economy") subsection of the Nøgletal ("key figures") section (pages 37-38), the table lists

Bloktilskud 3.722,4 kr. (2017) og 3.822,9 kr. (2018) millioner i tilskud fra Danmark

"Bloktilskud" means "block grants/subsidies", so that in 2018, Denmark gave Greenland 3.8229 billion Danish kroner in aid. Based on current exchange rates, this comes out to $553 million (United States dollars).

$553 million is close enough to $700 million that Trump isn't too far off. So Denmark does subsidize Greenland - but to the tune of about $553 million per year, not $700 million. Moreover, the government statistics do not imply that Denmark is "carrying" Greenland; the total subsidy is equivalent to only 25.5% of Greenland's GNP of $2.17 billion.

In terms of imports and exports, Greenland imported $787 million in 2018, exporting $603 million. The primary trading partners are Denmark and Sweden, although the rest of the EU also trades with Greenland. Denmark's relationship with Greenland, therefore, may be beneficial from a trading standpoint, gaining it money compared to a scenario where the US (or another country) owns Greenland. This is another thing that has to be considered when determining what Denmark gets out of the present setup.

That said, Trump's implication, of course, is that this subsidy is a waste of money for the Danish government. This is nonsensical, as Denmark is merely paying money to maintain an asset. Now, valuing that asset is extremely difficult; the Washington Post writes

So when The Washington Post asked the experts at the Arctic Institute to take a crack, Marc Jacobson, a senior fellow there, put it bluntly: “I’m not aware of any who would be capable” of doing that sort of calculation.

The Post tried several different ways of evaluating what Greenland is worth, and came up with values between $200 million and $1.7 trillion. The latter figure is close to an estimate by Jason Barr based on putative values of Greenland's land. However, land isn't what makes Greenland valuable; instead, its resources do. A fairly comprehensive 2014 report by the Brookings Institute reached several conclusions:

• There is an unknown amount of oil in around Greenland, although exploitation of that is years in the future.


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  The same holds for precious metals (gold, iron, rare earth elements, etc.):

  
  
  

  
  

  Greenland is widely believed to hold excellent potential for a host of natural resources, including zinc, lead, gold, iron ore, heavy and light rare earth elements, copper and oil. Considering that only a small fraction of this massive island has been properly explored, in the coming years more data gathering and analysis would be helpful to assess the full potential of Greenland.

  
  

  
  


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  Large-scale mining is possible in Greenland, especially if Chinese companies take interest.

  
  
  

  
  

  . . . because of the slowdown in investments in new mining activities, it is less certain that Greenland will be able to get major mining projects off the ground on the ambitious schedule that it announced in its mineral and energy resource strategy in 2014. We do believe however that eventually large-scale mining will take place in Greenland.

  
  

  
  

A true estimate of the value of Greenland is impossible until we can determine whether these future developments will occur and how much natural resources Greenland holds. New commodities like river sand are opening up for trade. The crux of the Brookings report is that Greenland has the potential to make a lot of money in the future. Even if Denmark is, at this minute, losing money by owning Greenland, it is impossible to say whether it will reap many more benefits in the decades to come, if development opens up.

Saying that Denmark is carrying Greenland is like saying that $50 a year renting a safe deposit box is carrying a small metal box that could be bought with a one-time payment of less than that.

Denmark isn't subsidizing a small community of 50,000 people; it is paying for the storage of its wealth.

Greenland contains about 7% of the world's fresh water.
How much is that worth?
With increasing population and pollution, how much will that resource be worth in the future?

How much petroleum is under that ice?
How large are the mineral deposits there?

Saying that Denmark is carrying Greenland is true only in the technical sense of "carrying charges" (an expense or effective cost arising from unproductive assets such as stored goods or unoccupied premises).

According to this BBC article from the time of the independence vote, http://news.bbc.co.uk/2/hi/europe/7752660.stm the "block grant" subsidy was fixed at DKK 3.2 billion / year. At today's exchange rate, that's about USD 480 mllion. It is somewhat less than 700.