Is there any site which shows historical air ticket prices between two pairs of places?

Yes, FlightAware Insight allows you to see minimum, median, and maximum prices on a given route over the last year, but it currently only has data for the USA.

For example, searching for flights from Nashville (BNA/KBNA) to Atlanta (ATL/KATL) yields these results:

According to FlightAware's FAQ, this data is provided by the airlines every 30-90 days, but is currently only available in the United States.

What is the coverage area of airline insight data?

FlightAware Insight currently cover airline and cargo operations in the US.

How current is FlightAware's insight data?

FlightAware Insight data is updated every 30-90 days. FlightAware currently displays the last year of data.

How does FlightAware know the fare/routing ticket details?

This ticket data for flown itineraries is provided by the airlines on a regular basis but with personal information (e.g., name, frequent flyer number, address, method of payment) removed.

Note on Ticket Prices

When looking at these prices, keep in mind that a large percentage of travelers on many routes (if not most routes) are business travelers buying expensive last-minute fares, not casual travelers booking cheap fares 4 months out. As such, the median prices you see here will probably be higher than you expect if you fall into the latter category and aren't necessarily good guidance about what constitutes a 'good' price for the route. The minimum fares work better for that purpose, though.

Note on "Airport Code" Search Fields

The "airport code" they're looking for in the search fields on this site is the ICAO identifier for the airport (used by pilots and ATC,) not the IATA airport code (used by airlines for ticketing and baggage handling.) Thankfully, converting between the two is easy for the USA. For airports in the continental 48 states, in general, just adding 'K' to the front of its IATA code will yield its ICAO code. For airports in Alaska or Hawaii, add 'P' instead. Unfortunately, it's not so easy for airports outside of the USA, where there's often no correlation at all between the two codes, so you just have to look up the ICAO code.

I agree with @chx (for reasons expressed in a different way here) but go further and suggest the data would be useless for your purposes anyway. When fuel prices dropped 70% fares to Spain went up 10%. Others fares did fall but by significantly different amounts. There are far too many other factors involved (exchange rates, competition, demand, improved fuel efficiency, etc) for even a change over one year in what comprises less than 30% of ticket price to have a very noticeable impact, even if of immediate effect.

And another issue is that the impact is not immediate. Most airlines hedge† fuel costs and buy in the future at prices set in the present. Unexpected changes in the commodity price may have no impact for six months or more. Even the few that do not hedge normally cannot, or chose not to, pass through input price changes directly and immediately into prices because they are often in fierce competition with others that do not pass on such changes immediately.

† considered at How do oil prices impact flight prices?

Of course it is near impossible to prove a negative but I will venture a historical database of all airline fares does not exist. You would need to have an agreement with all the Global Distribution Systems (but at least the large ones) to get a "ping" every time a fare is entered or a ticket is sold or any other relevant change happens to any airline fare. Where's the business in this for the GDS? Oh and if you were able to get hold of this data, you might have a chance at playing the on demand pricing game better simply because you'd have more data than the airlines themselves.

Noone has any incentive to sell you such a feed.