The embattled attorney general resigned at the "request" of Trump on Wednesday. Sessions' chief of staff, Matthew Whitaker, will serve as acting attorney general. Whitaker will assume oversight of special counsel Robert Mueller's probe into Russian election interference and possible collusion by Trump's campaign into that meddling.
Whitaker, who has publicly criticized the Mueller investigation, will have the power to fire the special counsel "for cause."
Rep. Nancy Pelosi, D-Calif., said Sessions' "firing" was a "blatant" attempt to undermine the probe. Pelosi is expected to become the speaker of the House in January.
Rep. Jerry Nadler, D-N.Y., the anticipated House Judiciary chairman come next year, called for answers. He tweeted Wednesday afternoon: "We will be holding people accountable."
Dan Clifton, head of policy research at Strategas Research, said he "absolutely" thinks Democrats will work with Republicans to pass bipartisan legislation despite possible investigations.
"It's about timing. We're going to have to pass a budget. We're going to have to raise the debt ceiling. We're forced into that regardless of what's happening in the investigations," he said on Closing Bell."
"At the end of the day, the Democrats are going to have to bring results home so that they can win re-election. They won very tight races in Republican districts to take over the majority."
What interests Ed Mills, policy analyst at Raymond James, is the escalating trade tensions between the U.S. and China. And he's concerned that a possible investigation into Trump may ramp up that rhetoric, he said.
"Trump has a history of whenever there is a series of negative headlines to try to change the subject," he told "Closing Bell." "The one subject that he can absolutely change the press narrative on is doubling down on China."
He also said if Trump ultimately winds up firing Mueller, the market may brush it off.
"One of the things that we see with the Trump administration is that something that seems like the biggest thing today, because Trump goes from crisis to crisis to crisis — we generally move on," Mills said.
"What the market will look at is the deregulatory agenda that is coming out of the Senate and what could actually happen legislatively out of the House. … Anything that's even potentially on that list is more fiscal stimulus."
— CNBC's Jacob Pramuk and Dan Mangan contributed to this report.