Running in 2024
Neither a second House impeachment nor even a Senate vote to convict Trump and remove him from office would prevent him from running again, in 2024 or beyond.
Rather, after two-thirds of senators present voted to remove Trump, a simple majority of senators present would have to approve an additional vote to bar him from the presidency in the future.

There is at least some uncertainty about the disqualification issue, since no president has ever been removed from office by the Senate and only judges have been disqualified from future office. The disqualification language in the Constitution is "disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States";