Reference for annual journal subscription costs paid per university?

There is, in fact, a resource with the information you asked for, for institutions in the United States.

Detailed information on individual academic libraries' expenditures (by university) is available from the National Center for Education Statistics in the United States, as part of their Libraries Statistics Program.

The data from these surveys, including the individual responses from each university, are available for download in plaintext format at this link. The most recent year available is 2012.

To take your example, according to that data, Harvard's expenditures for library resources in 2008 included $9,248,115 for serial subscriptions. In 2012, this number was up to $16,391,638 (the most of any library in the survey).

If you're interested, a set of related information on amounts paid by selected public universities to specific major publishers can be found in

Bergstrom, Theodore C., et al. "Evaluating big deal journal bundles." Proceedings of the National Academy of Sciences 111.26 (2014): 9425-9430.

(see especially the supporting information for the latter.)

I know you're primarily interested in the US, but others reading the question title may be interested in other countries. For the UK, see

• Wiley/Springer/T&F/Sage/OUP/CUP (collected by Stuart Lawson and Ben Meghreblian)
• Elsevier (collected by Tim Gowers)

For some information about New Zealand universities, see Mark Wilson's blog post and spreadsheet.

I don't know for sure whether there's a database that shows total journal spending at a fairly comprehensive list of universities, but I doubt there is. There's not as much transparency around university budgets as one might hope (especially at private universities, but even at public universities). [EDIT: As ff524 found, I was wrong about the existence of such a database.]

There is certainly no large-scale database that breaks down library budgets to show the amount spent for each publisher. In fact, many bundle contracts explicitly keep this information secret. Taking advantage of open records laws, Bergstrom, Courant, McAfee, and Williams managed to collect information about a number of contracts with public universities, leading to a paper and further information. However, there are still many universities at which this data is officially confidential.