| Sometimes. It depends on the design of the DI, and what you're trying to achieve. For example, a DI typically has two outputs, the DI out (meant to feed a mixer or a mic preamp) and a "thru" out meant to run into the instrument input of your stage amp. You can put effects before the DI if you want their sound to go to the mixer; or you can put the effects between the "thru" out and the stage amp if you don't want their sound in the PA or the recorded track (yes, there are some very good reasons for doing that). The "thru" output of a DI may be a direct uncolored, unchanged connection to the input signal; or it may be after any tube, transformer, or other coloration circuit stage, with all tonal qualities included.
IOW, it depends. | 
