LEDs do produce heat, but not nearly as much as a related incandescent. The LED is a semiconductor which may be prone to a phenomenon called thermal runaway. The movement of electrons (and holes) through the PN junction of the diode results in heat. This heat increases the conductive efficiency of the junction. As a result, unshed heat can result in a cascade reaction which eventually destroys the junction.
High-power semiconductor devices shed a lot of heat. Take a gander at the heat sink in your desktop computer. Look then at the size of the package in which the processor is contained. Consider that the processor itself is about a centimeter square within that packaging. LEDs are no different in this regard. The purpose of the heat sink is to cool the junction. I've touched the heat sink of an operating LED bulb, and found it to be warm to the touch. The wattage rating (real, not effective tungsten output) should give you an idea of how much heat the device will shed into the atmosphere.
We don't use standard dimmers with some bulbs because of the nature of the power supply circuitry—it probably requires a certain line voltage in order to properly function. Low voltage conditions (i.e. standard wall dimmers) may not be appropriately handled by the power supplies. I would imagine that, for much the same reason, the voltage transient which occurs when relays switch may cause difficulty for some units.
The worst case scenario is that the diode itself would be destroyed. Whatever the case, though, you should be able to find LED units which can handle dimmers and relays. As with anything else, you get what you pay for.