Let's start with the most important point first:
A campaign finance violation is not a ground to remove an elected official from office, no matter how egregious, on its own, even if one could prove that the campaign finance violation probably caused the outcome of an election to change.
Congress could decide, however, that a campaign finance violation constitutes a "high crime or misdemeanor" for which a sitting President could be impeached, if a majority of the House of Representatives votes to impeach the President and a two-thirds majority of the U.S. Senate trying the impeachment under the supervision of the Chief Justice of the United States (that and not "Chief Justice of the United States Supreme Court" is the official title) finds that the offense set forth in the impeachment has been committed and constitutes sufficient grounds to remove the President from office.
At the outset, there is some ambiguity over whether the violation of campaign finance laws would have been by individual person, or the political campaign of the individual person (which is a Section 529 non-profit organization).
The individual or the campaign, as the case might be, would definitely owe either civil or criminal fines, as the case might be, for a violation of the particular campaign finance laws that the individual or campaign is alleged to have violated. The violation could also affect the ability of the violator to obtain all or any amount of federal matching funds in a future election.
Depending upon the particular campaign finance law involved, which isn't clearly specified, there might be a criminal penalty (either a misdemeanor or a minor felony) as well, which could give rise to a probation sentence, a sentence to some period of incarceration, a period of post-release parole, criminal fines, court costs, and, at least in the case of a felony, some collateral consequences of that conviction such as a prohibition on owning a gun and ineligibility for many occupations and jobs.
Neither a civil campaign finance violation, nor a criminal conviction for a campaign finance violation, disqualifies someone from holding the office of President.
The President does not have immunity from civil or criminal consequences of campaign finance violations committed prior to taking office, and the President's campaign, at a minimum, is not immune to campaign finance violations at any time. Presidential immunity from liability is limited to the conduct of the President while holding the office in furtherance of the governance duties of the President.
Generally speaking, campaigning would not be a governance duty of the President, but there are factual circumstances under which it is hard to distinguish between what constitutes governing and what constitutes campaigning in the case of a sitting President who is currently in office.
Also, campaign finance violations can't be prosecuted by just anyone. A violation of federal campaign finance laws must be prosecuted by a federal government lawyer authorized to do so by the Federal Election Commission (FEC), which has an even partisan balance of members by design and almost always deadlocks on motions to prosecute anything but the least controversial and most blatant campaign finance violations against members of either of the two major political parties in the United States. A "private attorney general" can't bring a lawsuit to enforce campaign finance laws unilaterally.
The President can definitely pardon a criminal violation of a campaign finance law by anyone other than himself. The President definitely cannot pardon a civil violation of a campaign finance law by anyone. The majority and more analytically sound position is that the President cannot pardon himself for his own criminal violation of a campaign finance law (there are prior questions and answer at Law.SE regarding that question).
A judge considering a campaign finance violation charge against a sitting President could, however, take steps short of dismissing the charges that could accommodate a sitting President.
For example, the judge could be very deferential to the President in setting hearing dates, allowing the President to participate in proceedings by telephone or (in some but not all cases) through a representative, or in setting the amount of a bond or the terms of pre-trial release in the event of a criminal charge, or in cooperating with the President's security detail when the President is required to appear. In a civil violation case, the President would probably be allowed to be a deposition witness, with the deposition testimony used at trial, rather than appearing in person at trial, if his lawyers requested that treatment.
