Considerations when repairing dimmer switch
To test the triac out-of-circuit you can use an ohmmeter on the lowest range. You need to find the datasheet. If it's the type with a built-in diac you won't be able to easily test the gate with low voltage. 

Test for continuity between MT1 and MT2, in both directions. 



If you find continuity in either (or both) direction(s) then the triac is knackered, end of story. This is the most likely failure mode. It is typically caused by an overcurrent surge even such as arcing during the failure of an incandescent bulb. 

Another test that can be done is to hold the meter terminals (using the lowest ohms range) on MT1 and MT2 and momentarily touch MT2 to gate without breaking either of the connections. The meter should show continuity as long as the gate is touching, and may continue to show continuity after the the gate connection is broken. You can test it both ways. 

A more functional test that is a bit more trouble to set up is to get a 12VAC transformer and a 12V few-watt light bulb (maybe an automotive bulb) and connect the triac in series with the bulb to the transformer secondary. Use a 100 ohm resistor to momentarily connect gate to MT2. The light should go on as long as the gate is connected. If the light does not turn on or the resistor gets really hot the triac is bad. If the light goes on without the resistor (full or partial brightness) then the triac is dead. 

Personally, I would order the triac immediately without testing. There is no other plausible failure that will cause the light to remain on at full brightness, and the triac failure is likely the only failure.