May the Senate convict against the evidence? Similarly, the answer here is also no. If anything, this answer is even clearer than with the House. Senators swear an oath for impeachment trials to “do impartial justice according to the Constitution and laws.” If a senator concludes that the president’s conduct, as shown by the evidence at trial, does not amount to an impeachable offense, they are honor-bound to vote for acquittal. Again, conviction on any other basis would render meaningless the Constitution’s specification that officials shall be removed only upon conviction of a high crime or misdemeanor.
The power of impeachment and removal is limited and enumerated, like the rest of Congress’s powers, and if either the House or the Senate transgresses those constitutional limits, it has violated the law. True, there is no remedy for that violation, but that is all the more reason why Congress itself must take seriously the constraints on impeachment, and act against the president only on the basis of a sincere, considered legal judgment.
May the House decline to impeach against the evidence? This is where things get interesting. Can one imagine a situation in which the House should not impeach the president, even with a clear legal basis to do so? I think so. Suppose, for example, that during a time of war the president was revealed to have taken cash bribes from the enemy and deliberately, but surreptitiously, undermined the war effort, resulting in the death of American soldiers. Treason and bribery are enumerated impeachable offenses; assuming that the facts were clear, this is about as easy a case of impeachment as could be imagined. But now imagine that these facts come to light only a week before the president's term is due to expire, after losing reelection. It is hard to say that the House would be under any sort of obligation to impeach, rather than waiting for the president to leave office naturally.
This scenario is extreme in both directions, but it demonstrates that the House may properly take into account these sorts of prudential considerations. This accords with the House’s role in the process, which is often compared to that of a grand jury, but which is also analogous to that of a prosecutor. (For example, on the one hand, the House appoints the impeachment managers, who present the case for removal to the Senate. A criminal grand jury, on the other hand, does not choose who will present the case against a defendant at trial.) And prosecutors routinely exercise discretion about whether to pursue cases even if there is enough evidence to convict. Nor is timing the only valid ground on which the House can exercise its prosecutorial discretion. Imagine, for instance, a president who courageously decided to expose their own misconduct in the face of attempted blackmail. The misconduct might well be gross enough to warrant impeachment, considered in a vacuum, and yet the House might feel that the president should be rewarded for showing integrity and placing the national interest ahead of their own. Or perhaps, in a marginal case, the House might simply conclude that impeachment is not worth its disruptive impact on the nation.