“My loan officer has told me that on a 5/1 ARM, paying $1000 more each month for 60 months won't generate any savings over paying $60,000 in month 60, just before the interest rate and payment reset. Is this true?”  

No, he is wrong. If you pay $1,000 a month for 60 months, the loan balance will be smaller in month 60 than if you applied $60,000 to the balance in month 60. The reason is that extra payments received early reduce the balance early, which reduces the monthly interest due on all future payments, which increases the portion of all future payments that is applied to principal.  

The higher the interest rate, the larger the savings. For example, at 5%, paying an extra $1,000 a month on a $300,000 loan will result in a loan balance 3.7% smaller after 60 months than applying $60,000 to the balance in month 60. At 10%, the balance would be 7.6% lower.  

This principle holds whether the loan is an ARM or an FRM, whether it is a standard monthly accrual mortgage or a simple interest (daily accrual) mortgage, or whether the scheduled payment is fully amortizing or interest only. The only kind of mortgage on which it would not hold is one on which the loan contract prevents the borrower from making extra payments, and to my knowledge there are no such home mortgages in the US.