Not that I'm an amp or audio expert - refer to others here else the12volt.com.
But the remotes I have seen are mere low-current inputs that energise the amp's main power. Usually they energise a relay direct though some might energise electronic circuits that energise the relay (or solid-state relay else "kick" the amp's PSU into operation).
Hence they should be from mA to maybe 2A (assuming the man power is connected to other terminals).
And hence too how fusing probably won't protect the amp's remote - it's like fusing a LED or relay coil - pointless. EXCEPT to protect the device powering the remote (or LED or coil etc)...
IE - If the thing supplying the remote can only handle 1A, then a 1A or smaller fuse to protect it and its 1A or greater rated wire to the remote in case that wire shorts to GND, or the amp's remote has some internal short or fault.
IOW the fuse is then merely the normal consideration - protection of the source and its distribution (wire etc).
As to the comment "don't worry .... the product will only draw what it needs" - yeah, a frigging useless comment, and self apparent to anyone with a bit of knowledge. (Hey man, don't fuse anything because they only take what they take - including when they short!)
YTF can't manufacturer's supply the SIMPLE detail - the remote draws 450mA for its relay coil, or 15mA for its transistor circuit, etc? And stuff like "it's protected from external relay spikes and surges" or +/- 100V inputs etc?
[And some wonder why they go out of business or lose sales to elsewhere!]
If the remote itself is the main power feed, then that's a different kettle of fish. But them IMO it is not a "remote", it is the "power" input. And all things tied to any input are "remote" if anyone wants to use that argument.
And be aware, I am aware of cases where the remote does draw lots of power, especially when the main power fails. But IMO that's the result of a fault. (I think one used solid-state inputs and relays and it probably failed in short-circuit mode (ie, low current paths ties direct to high-current rails); maybe a Sony amp?)