Is there any legal precedent for suing a city to amend or terminate an
agreement due to fiscal nonfeasance?
There is not really any legal precedent for prevailing in such a lawsuit.
Obviously, of course, the detailed facts and circumstances matter. If a state statute prescribed other terms, for example, and expressly gives someone standing to enforce the statute, then that is another matter. In many states, standing to enforce violations of municipal laws governing their finances and contracts is vested by statute or the state constitution in the state attorney general.
Is there any legal recourse for a resident who believes their city is
committing financial nonfeasance?
Probably not. Certainly not in court.
Usually, individual citizens or taxpayers do not have standing to bring suit related to acts which affect all citizens or taxpayers equally or proportionately, but do not constitute an individualized injury to the particular taxpayer.
Municipal governments have broad discretion to enter into contracts with other municipalities on rates that they deem fit which do not have to approximate cost or be profit maximizing.
Some states and cities allow citizens to petition to have legislation that has been adopted (agreements are generally adopted by city ordinance) to be placed on the ballot for a vote if a sufficient number of people vote on it within a sufficient time of the ordinance or law being passed (this is called a "referendum power"). But, most do not.
Otherwise, your sole recourse is to get a majority elected to city council and a new mayor, to change the policy when the agreement expires.
wouldn't the city have to prove that there IS a benefit to the city?
Assuming for sake of argument that someone suing the city had standing to sue, the burden of proof is always on the person bringing the lawsuit. Ordinances are presumed valid unless this is disproven beyond a reasonable doubt. For example:
It is an axiom of our judicial system that legislative enactments are
presumed to be constitutional. Parties attacking their validity carry
a heavy burden of proof: invalidity must be established clearly and
beyond a reasonable doubt
People v. Beaver, 549 P.2d 1315, 1316 (Colo. 1976)
The constitutional test in the face of an equal protection challenge (assuming for sake of argument that there was standing) would be a "rational basis test" and there would be a rational basis for (1) saying that the city benefit from its neighbor not having adequate fire protection which could spread to them, (2) on the basis that the marginal cost might be low, and (3) on the basis that the municipality probably has a legal duty to aid a neighboring municipality if it has the ability to do so in the absence of an agreement without necessarily having a right to compensation under a doctrine called mutual aid when the proper conditions are met (sometimes formalized by agreements and/or governed by state statutes such as the Tennessee's Mutual Aid and Emergency and Disaster Assistance Agreement Act of 2004, Tennessee Code Annotated § 58–8–101, et seq.,).
The rational basis test is met if you can describe some rational reason why the law might make sense for the city to pass (which is not expressly prohibited by law or a constitutional right), even if the rational reason is not empirically correct, and even if the rational reason wasn't the actual reason for passing the law. This ordinance would almost certainly pass the rational basis test.
In general, a disagreement over the price term of an agreement being too high or too low almost always fails. A municipality is not obligated to negotiate a "fair market value" or "fair" price for services that it provides to other municipalities.