(1) In this paper (published J. Combinatorial Designs, 15 (2007) 98-119), in the history section starting page 3, we cite many published errors in counting Latin squares and related objects. Some, but not all, were before the computer age but required substantial hand computation.

(2) The number of closed knight's tours on a standard chessboard was first published here. The answer is in the title of the paper, but is unfortunately incorrect. See the comment there for more information — the authors later replicated my answer so it is presumably correct.

Of course programming errors and clerical errors (e.g. putting the results of multiple computer runs together incorrectly) are the main cause of published errors, but hardware errors also occur. I've had individual computers in clusters of "identical" computers that regularly gave answers that looked perfectly reasonable but were wrong.

In the early days of silicon memory, the most common error was due to alpha particles from impurities in the silicon. Then, as silicon purification became more advanced, cosmic rays became the major factor for memory errors. Now I think that the main problem is that the components are so tiny that random noise and cross-induction are key. Also, memory with error-correction is more expensive than memory without so usually only high-end computers have it.