Can an employer fire an employee who is eligible for retirement so they prevent the employee from receiving retirement benefits?

An employer cannot fire someone simply to prevent them from getting retirement benefits; but they can fire a retiring employee for other good cause, and if someone is properly being fired, they don't have to let him or her retire (or, for that matter, resign or quit) first.
It depends on the timing and facts, and every case is different. To give some examples:
1) You announced you were retiring first, before taking the test: they can't then fire you without letting you go through with the previously announced retirement (unless you do something justifying "for cause" firing, like excessive absenteeism, open insurbordination, falsying reports or time sheets, etc.).
2) They do not normally fire people who cannot or do not complete promotion training--in that case, it was likely improper to terminate the older employee when younger ones were not terminated.
But:
3) If they do normally terminate people who fail this promotion training, then they could terminate you, too, if you had previously (prior to failing) announced retirement.
4) If you did something that would in and of itself justify termination without regard to age--for example, the reason you failed was that you did not attend training sessions, or you were insubordinate--termination is justified.
If you feel that based on the timing and circumstances there was no good cause for your termination and they terminated you, an over-40 employee, simply to prevent you from getting retirement benefits, then you should contact the federal EEOC or your state equal/civil rights agency to discuss the matter and possibly file a complaint. Good luck.