I want to replace a dimmer which that operates overhead incandescent bulbs (each 25 watts) totaling, as I remember 725 watts so I expect to use a 1000 watt dimmer. Here is my issue--the circuit that this switch is on has a 20 AMP breaker. Do I need to find a 20 amp dimmer switch for 1000 watts or can I use a 15 amp dimmer for 1000 watts? I cannot seem to find a 20 amp dimmer switch for 1000 watts.

The dimmer will work if it is rated for the load it will control. The amperage of the supply doesn't matter in this particular instance.

Do I need to find a 20 amp dimmer switch for 1000 watts or can I use a 15 amp dimmer for 1000 watts? I cannot seem to find a 20 amp dimmer switch for 1000 watts.
There is no such thing as a 15 or 20 amp dimmer. Dimmers are rated in watts only.

Closest thing I've seen is a 2000 watt dimmer. At least as far as dimmers that will install in a normal outlet box.