The MLB Draft is coming up tonight and will last the next couple of days. Throughout that entire time, there will not be a single trade done involving any picks or selected players. This is because trades are prohibitted by the league.
The MLB Draft is the lone draft that doesn't have some sort of trading. In the NBA Draft, rights to drafted players are traded all the time. NFL teams swap picks ad nauseam. Why aren't MLB teams allowed to do so?
Why MLB Draft picks cannot be traded
Standard picks in the MLB Draft cannot be traded, though compensatory picks (often awarded to small market, poorer teams) can be. In general, this doesn't occur.
The reason for this is competitive balance. In a sport without a salary cap, teams can theoretically spend as much as they want. This creates a divide between the "rich" teams and the "poor" ones, though MLB owners can spend as much as they want.
The draft trading prohibition was enacted to prevent the gap from widening. Say the Pittsburgh Pirates don't want to or can't afford to pay Dylan Crews, who wants $10 million. They can trade the pick to the New York Yankees, New York Mets or Los Angeles Dodgers.
Those teams can afford him and the Pirates would recoup a lot of capital in return. Ultimately, given how hit or miss MLB prospects are, this would keep the Pirates bad and make the rich teams better. The league doesn't want that.