With tmux 1.5, the capture-pane command accepts -S and -E to specify the start and end lines of the capture; negative values can be used to specify lines from the history. Once you have the data in a buffer, you can save it with save-buffer.
Here is an example binding (suitable for .tmux.conf) that wraps it all up with a prompt for the filename:
bind-key P command-prompt -p 'save history to filename:' -I '~/tmux.history' 'capture-pane -S -32768 ; save-buffer %1 ; delete-buffer'
This captures (up to) 32768 lines of history plus the currently displayed lines. Starting with tmux 1.6, you can use numbers down to INT_MIN if your pane has a history that is deeper than 32Ki lines (usually up to 2Gi lines). Starting in tmux 2.0, you can use capture-pane -S - to mean “start at the beginning of history” (i.e. no large, hard-coded negative number).
Note: The number of lines in the saved file will not always be equal to the pane’s history limit plus its height.
When a pane’s history buffer is full, tmux discards the oldest 10% of the lines instead of discarding just one line. This means a pane’s effective history depth will sometimes be as low as 90% of its configured limit.