"I think this is definitely a bigger deal than some of the other departures, due to Cohn's reputation as proponent of Wall Street, said Dan Deming, managing director at KKM Financial. "Short-term I believe it would increase concerns of an increased probability of a trade war escalating."
However, he and other analysts noted that it was too early to determine whether there will be lasting damage to market psychology.
The U.S. dollar index extended losses, trading 0.5 percent lower near 89.56. Gold futures were trading near $1,333.60.
U.S. stocks closed higher Tuesday, recovering from earlier concerns about a possible trade war.
The major averages declined earlier in Tuesday's session after Bloomberg News reported, citing sources, that Trump was convinced Cohn would leave the administration if the tariffs proposed by the president were implemented.
"The statement was, if Trump went forward with these tariffs, Cohn would leave. It's tempting to think that since Cohn is leaving, they're definitely coming through. … I think that's a bit premature," said Nathan Sheets, chief economist at PGIM Fixed Income.
"At a minimum, though, this is a powerful voice inside the White House that will not be there arguing against [tariffs]," Sheets said. "Therefore, the probability may be higher that they go through."
Later Wednesday, investors will be inundated with a raft of economic data and comments by Federal Reserve speakers.
On the economic front, mortgage applications are set to be released at 7 a.m., followed by the closely watched monthly private-sector jobs report from ADP at 8:15 a.m. Readings on fourth-quarter productivity and January international trade are due out at 8:30 a.m.
William Dudley, president and CEO of the Federal Reserve Bank of New York and Raphael Bostic, president and CEO of the Atlanta Fed, are both expected to speak later Wednesday.
Check out some of CNBC's coverage of markets and tariffs:
Trump's tariffs may anger China and here's how it may retaliate
Goldman Sachs: Trump's tariffs are 'draconian' and will raise prices
Trump steel tariffs hit raw nerve in energy: US pipelines for oil boom rely on foreign steel
Goldman: GM and Ford could take $1 billion hit each from the steel tariffs
— CNBC's Fred Imbert and Tom Franck contributed to this report.