President Trump was impeached during his time in office, but he was recently acquitted and President Clinton was impeached, but was able to remain in office. Are there any further political consequences the current president or any future president can face from impeachment even if they are acquitted/remain in office, or do the consequences of impeachment stop once the Senate decides not to remove the commander-and-chief (other than the stigma of being one of the few American leaders to have been impeached)?
We don't really know and it depends a lot what you mean by "consequences" too...
In two of the three impeachment cases that went almost this far, Andrew Johnson and Richard Nixon, the legislature in each case accused the president of abusing his powers and the legislature in subsequent years scaled back some of the president’s power. We didn’t see that after the Clinton impeachment, and I think one of the reasons is because 9/11 happened so quickly after, and it just changed the entire conversation overall. So typically, the legislature would recognize it’s their responsibility to encroach back on executive power because they’d want to make the legislature more powerful.
I don’t think that’s going to happen in this case, because legislators from the beginning have viewed this in partisan terms rather than constitutional terms.
Trump like Johnson (and unlike Clinton) is in his first term. But unlike Johnson it's extremely unlikely Trump will be dropped from the ticket...
But, here are the differences: Andrew Johnson was dropped from the presidential ticket in 1868 (the year of his impeachment was an election year). And the Republicans, the opposition party, knew whom they wanted to head their own ticket: war hero General Ulysses S. Grant, who could unite all ends of their party. In fact, it was the 500,000 votes of black men in the South that carried the election for Grant — the very men that Johnson had tried to deny the vote.
I.e. the past is a poor predictor of what the consequences might be in this case. We probably won't see a repeat of some past consequences in such cases.
Fivethirtyeight has a long article speculating what the consequences might be for Trump and the Democrats in the near future. It basically predicts that polarization will deepen.
As for the consequences for future presidents, (mostly) Republicans (e.g. Senator Joni Ernst and Trump's lawyers) have argued that Trump's impeachment makes future partisan impeachments more likely. Democrats had argued the same at the time of Clinton's impeachment. Cipollone actually replayed some clips to that effect during his defense of Trump.
No. Once the impeachment trial has ended and a Not Guilty verdict has been reached, that is the end of the impeachment process.
Barring any subsequent impeachments by the House of Representatives, President Trump will be able to continue to the end of his current term in January 2021, and may stand as a candidate in this November's election.