Nixon (oh wait, he actually was)
Richard Nixon wasn't impeached; he resigned. Bill Clinton actually was impeached, although not removed from office.
But this shows part of the issue. Nixon resigned and replaced with Gerald Ford, who then had less moral authority in dealing with Congress. That was somewhat relevant at the time because the Democrats controlled Congress (both House and Senate). Ford ran for election, but couldn't rally support behind himself. Consider what could have happened if Nixon had completed a normal second term. Someone like Ronald Reagan could have run to succeed him. Rather than running against an incumbent, Reagan would have had an open field. Reagan almost beat Ford as it was.
Another part of the issue is that they're trapped in their own rhetoric. If you argue that your opponent is of low moral character and does things that are illegal, then you have to argue for impeachment. If you don't suggest it, then someone else will. You have to agree, because otherwise you just sound silly. Such-and-such is really horrible, the worst ever, but not quite bad enough to impeach. See? Silly.
Finally, sometimes it does matter. If George W. Bush and Dick Cheney had been impeached in 2007, then Nancy Pelosi would have been president. In most modern administrations, the vice-president would be entangled in any actual wrongdoing, like use of torture.
If Nixon had been impeached after the resignation of Agnew, Carl Albert would have been president. They let Ford become vice-president as part of a compromise to remove Nixon.
If Obama had been impeached and Biden had become president, Republicans might have been able to accomplish more. For example, there was a rumor of a grand deficit-reduction bargain between Biden and Republicans that came to naught when Obama rejected it.
House Speaker Newt Gingrich, who, prior to the election, had been reassured by private polling that Clinton's scandal would result in the Republican Party gaining as many as thirty House seats.
The expectation at that time was that even a failed impeachment of Bill Clinton would improve the Republicans' position in the House and Senate. And in that case, the Republicans would almost certainly have been able win the presidency in 2000 (as they did anyway) and keep or even increase those gains. The reality was that they lost seats.
Something similar had happened previously. Democrats won the presidency in 1976 after the Nixon resignation, but then lost the next three presidential elections. Impeachment cut both ways. Nixon supporters weren't enthusiastic about Ford, but Jimmy Carter never built the support that he would need for reelection.
If Hillary Clinton had won in 2016, Republicans would be in great shape now. They could be looking forward to winning around nine Senate seats in the 2018 election. They could hardly improve their gubernatorial holdings, but they probably would have kept them. They would be the ones talking about the president's shady dealings with Russia and pouncing on her every misstep (possibly including her firing of James Comey). But they won and got Neil Gorsuch onto the Supreme Court and repealed a bunch of regulations.
Given Trump's popularity (or lack thereof), impeachment might actually help Republicans. Or perhaps the outrage would hold until 2020, as happened with Nixon. We don't have a lot of examples from which to extrapolate.
Politicians may believe that impeachment talk helps them more than it actually does. It rallies the base and brings in contributions. But it also rallies one's opponents and brings in their contributions. The thing is that most politicians can see the effect on themselves more directly than that on their opponents. Or to put this another way, most politicians are near-sighted idiots. They see the gains without counting the costs.