Hi Luke. When you first buy your collets, I am certain there will not be instructions to grease the working parts of the tool holding system. Think of a collet as a vital and integral part of the tool holding system. It is almost a component with an interference fit. It is not quite that but dialectric grease is usually silicone based. Silicone is slippery and improving the slip between the collet, the collet taper and the collet nut does not appear, to my mind, to be mechanically sound.
A router bit leaving the collet at 30,000 RPM is not an event which you would want to experience. Grease can only attract the swarf or sawdust spoil which you would definitely want to keep out of the collet taper. Silicone grease will help to offset the collet (keeping swarf in place) and wear it out quickly and keep waste materials sitting inside the collet taper, which have no business being there.
The collet nut and taper of my trim router is cleaned at every tool change with a soft brush and air blown through the parts. If I used grease of any kind, I would be making that cleaning job far harder than it already is. It is not a difficult task if it is carried out routinely at tool change time. There is no excessive build up of swarf or sawdust in the spline gaps of the collet or inside the collet nut or the collet taper.
I am not claiming that this is the only way to manage collet components and routers that hold onto the toolbit using a sytem of taper and collet nut to hold the toolbit, or even that it is correct. In an area of the tooling that involves a router and a toolbit, I would say that there is no place for grease of any kind. This is notwithstanding that some theorists talk about boundary effects and boundary lubricants like those containing Molybdenum Disulphide or Graphite.
There is a whole slew of information on this subject in internet land and the application of grease does not feature very much in collet care. If a regularly cleaned collet system starts acting up even when clean, that is the time to change the collet for improved performance. At every tool change, where I engage in a routine of collet spline, collet taper and collet retention nut cleaning, I inspect the collet for signs of damage or wear. I treat the collet as a consumable item and replace it when necessary. YMMV