Do LED dimmers work with non-LED bulbs?
Probably...but check the fine print
There are a number of different light technologies in use (e.g., incandescent, halogen, fluorescent, LED) and dimmers have evolved to handle each new type. But some are easier to dim than others. In general, if you have a light that is dimmable then if you use a new dimmer it will work with it as well as newer technology lights.
For example, this Lutron dimmer can work with:

Up to 150W of LED or CFL (provided the LED or CFL bulbs themselves are rated as dimmable - not all of them are)
Up to 600W of incandescent or halogen

You might wonder why fewer Watts of LED &amp; CFL bulbs work than incandescent or halogen. There are technical reasons, but it doesn't matter because 150W of LED is comparable to 600W (or more!) of incandescent lighting - i.e., with any technology enough to light up a large room.
On the other hand, this less expensive dimmer from the same manufacturer/product line can handle 600W of incandescent/halogen but clearly states that it is not LED or CFL.
So the new dimmers are designed to handle the old light bulbs as well. Otherwise there would be a lot of problems as people upgrade dimmers without replacing the light bulbs at the same time. But plenty of people find out the hard way that you can't necessarily put a new bulb (LED or CFL) on an old dimmer.