Who is losing in an arbitrage? When some entity takes advantage of an arbitrage opportunity, who is losing money? For example, when there are price differences across cryptocurrency exchanges and someone exploits an arbitrage opportunity, who is losing out?

A vaguer question: what are the microeconomic implications of arbitrage?
Nobody has to loose in an arbitrage. Economic relationships are not necessarily zero-sum (in fact often they will not be zero-sum). For example, if apples in city A are sold for ${\\\$}5$ and apples in city B can be sold for ${\\\$}8$, and we assume zero transaction cost there will be an arbitrage opportunity to earn ${\\\$}3$ riskless profit per apple by buying apples in city A and selling them in city B. But nobody looses in the transaction.
The apple producers in city A clearly value ${\\\$}5$ more (or at least indifferent) than an apple otherwise they would not trade it and just keep the apple. So giving them ${\\\$}5$ will make them better off (or at least not worse off). Then in city B people who buy apples must value apples more than ${\\\$}8$ they are paying for it otherwise they would just keep the money instead of buying them (or at least they must be again indifferent).
Consequently, when people take advantage of arbitrage opportunities generally speaking nobody looses anything. There must be some other issues present that would turn the problem into zero-sum game.
Regarding the microeconomic implications the main one is that people will take advantage of the arbitrage opportunities until the prices on the markets equalize - in the example above until the prices in both cities would become equal. The law of one price is based on the concept of arbitrage. There might be other implications depending on precise setting where arbitrage occurs, to explore all of them would be beyond the scope of SE answer.