Like any good band, their influences showed early on and then were combined into a unique DT sound. However, DT has readily admitted to doing some songs or parts of songs as overt nods to their influences.
The bass line at the beginning of Home is apparently very reminiscent of Tool, and there's a Rush/Freewill section in Dance of Eternity.
Solitary Shell sounds to me VERY much like Rabin era Yes, and the guitar intro to About to Crash Reprise has a clear Styx vibe.
The synth solo in Octavarium and the solo in Along for the Ride both are nods to Keith Emerson
As mentioned, Looking Glass is an intentionally and overtly "Rushy" sounding song, and I think the bass part in the Pursuit of Truth section of Illumination Theory is reminiscent of Cygnus X-1, Book I.