Consider american options on interest rate futures such as the 10-year treasury note. When is early exercise optimal?
As OracleOfNJ said, there is never any advantage to early exercise of an American style call option unless the underlying asset offers some advantage, usually dividends, which does not apply to interest rate futures.
American put options were among the biggest open problems in finance until people learned how to treat them as free boundary problems. In other words, you'd first derive the Black-Scholes-like PDE that describes the equivalent European option product, but you'd solve it with a lower boundary condition described in terms of the PDE's solution.
I've never thought about American put options on interest rate futures specifically, but a Google search for "American put free boundary problem" yields some reasonable starting points, and an arxiv.org search for "American put" has many relevant articles.
I have a different take on this. Listed options such as options on the 10yr Treasury future, are NOT subject to daily variation margin. That is different to the underlying futures contract, which IS subject to daily variation margin. Let's say a listed option is 10 points in the money, with a month to go. If you do not exercise, you have an asset worth 10 points at expiration which you have to fund for a month at the short rate, so it's worth slightly less than 10 points. If you exercise early, you get a futures contract and then the next day you get 10 points of variation margin delivered to your account, so it's better. The only thing you have given up is the 10 point out of the money put due to your early exercise. So the rule of thumb is:
Exercise early if : (short rate * daycount * intrinsic) > put option given up