This question had only 1 answer prior to 2021-01-06, and circumstances changed on that day.
There have also been related cases since, and more things that have come to light.
Yes, Donald Trump now could be prevented from taking office, but it would require several things which are not, at this point, givens, and it relies on a different law to Amendment 14 § 3, and that law might not result in disqualification.
There is, indeed, an organization (C.R.E.W.) that is threatening suit if Trump is actually elected to federal office, but its legal argument does not hold water.
It's based on the wrong law, lacks evidence, and the right law would require federal criminal prosecution not a civil suit by C.R.E.W..
Moreover, it would not be the first time that a suit by this organization against Trump did not hold up in court.
Amendment 14 § 3 only talks about engaging in insurrection.
But Trump definitely did not do the necessary actions that constitute engaging in insurrection.
He famously did not accompany the marchers to the Capitol himself.
18 USC §2383 covers incitement to insurrection, and includes a prohibition against holding federal office upon conviction of that.
But, as I explained in an answer at https://law.stackexchange.com/a/60042/18361, the case that Donald J. Trump actually incited insurrection is weak, as there is no direct statement by him that did so.
(I actually went through the transcript of the speech myself before writing that answer.)
Telling people to "demand" that congresspeople "do the right thing" and "make your voices heard" is not, on its face, incitement to insurrection; because by such an interpretation any political activitist would be guilty of the same.
One could argue that there's a context for Trump's statements, but it is a weak argument that would hinge on (a) a reasonable person's interpretation of Trump's statements, (b) the crowd's interpretation, and (c) inference of Trump's intent when he did not actually say the necessary words.
Since I wrote that answer, Mike Pence has revealed, in an 2022 autobiography (So Help Me God, not yet published at the time of writing this answer) and in a Wall Street Journal article, that Trump asked him to disrupt the process of counting the Electoral College votes, several times over the preceding month and indeed on the day of the count itself.
This could potentially be a separate act of incitement.
However, it is again, not a given that Mike Pence's account of what someone else said is sufficient as evidence.
He has not (yet) given sworn testimony about this, let alone has such testimony been considered by a jury.
Furthermore, even if a jury were to give it credence, and even though it almost certainly qualifies as as incitement to something, another unsettled point is whether asking Mike Pence to do something necessarily qualifies as incitement to insurrection.
It was (by Pence's same account) on the advice of a lawyer who had told Donald Trump one thing and other people another, that said lawyer had told Donald Trump was legal but untried.
So there's question as to whether Donald Trump was knowingly inciting something that he had reason to believe was insurrection.
Again, it's not a given that such a case would succeed, if there's an untested counterargument that Trump honestly believed that he was encouraging something that his counsel had told him was legal, or even a counterargument (should someone claim that it is at least a felonious act to do what was being proposed) that soliciting the commission of a felony does not meet the standard of incitement to insurrection.
(Then there's the fact that 18 USC §373 requires that it be a violent felony that is solicited, which not counting Electoral College votes clearly cannot be, and does not result in a prohibition of holding federal office.)
There are a lot of things that are simply not givens in such a case.
Hence could be, not definitely will be.
As I wrote in that answer, there are far more clear cut cases against Eric Trump, Lara Trump, L. Lin Wood, and Rudy Guiliani; and should they ever try to hold office, a case against them has a far greater likelihood of success.
One can pass the "incitement to imminent lawless action" test (see this article for background) far more easily with people who directly say things like "trial by combat"; there is evidence that they used words that encompassed violent action.
Likewise, C.R.E.W.'s successful 2022 suit against Couy Griffin, erstwhile county commissioner in New Mexico, rested (per the decision) on his performing several explicit acts that met the Amendment 14 § 3 (and 18 USC §2383) requirement of engaging in insurrection; unlike the untested and weak claim under just 18 USC §2383 (and not Amendment 14 § 3) that Donald J. Trump incited it.
It was also a suit at state level under New Mexico state law.
So to reiterate, the relevant law is 18 USC §2383, which carries a prohibition on holding federal office, and allows for incitement.
Amendment 14 § 3 is only relevant to those who engage in insurrection.
And it isn't a given that incitement could be proven for Donald Trump, even though it could very likely be proven for other speakers, and even though engagement has been proven for others who climbed barricades.
Finally, 18 USC §2383 is a part of the U.S. federal criminal code, so the procedure would be criminal prosecution and conviction at the federal level. No impeachment would be involved, and it wouldn't be a state level suit by C.R.E.W. or a civil action.
And if you think that the aforegiven explanation, that you and C.R.E.W. are are looking at the wrong law and that there might not be the evidence to convict, is lengthy, just wait until you read this even longer argument that the President of the United States is not an "office under the United States" (written a month after my answer) and so even if it were to all hold up in court and result in a conviction for incitement to insurrection the outcome might not be disqualification to be elected President of the United States.
