While the articles of impeachment themselves would not bar Trump, one possible outcome of the impeachment proceedings is that he could be barred.
A president in the U.S. is impeached by the House and is subsequently convicted and removed from office by the Senate. This means an impeachment by the Democratic-controlled House does not alone result in Trump being barred from running for office. Even a Senate's conviction does not mean Trump will be barred; if the Senate did convict a president, a separate vote would be required to disqualify him from holding future office. Daniel Dale, a CNN reporter, tweeted: "According to Senate precedent, a post-conviction vote to prohibit the person from future office would require a simple majority, not the two-thirds that conviction itself requires."
We conclude that this claim is partly true because while strictly speaking it is not the articles of impeachment themselves that would bar Trump from holding further elected office, but rather a further vote following conviction in the Senate, it is nonetheless true that one possible outcome of impeachment proceedings is a bar to holding elected office.