While technically "consult a dictionary" is correct, I wouldn't use it.
Probably because having a consultation with an object vs a person sounds like one of the dumbest fucking things ever in the face of reality.
Like most Straw Man arguments, yours fails.
"Consultation" is a noun, whereas "consult" is a verb transitive. True, the first is derived from the second, yet if you were to consult your dictionary again you'd see that it can only denote an interchange between two people, not a person and an object. The notion of an interchange is central to its definition. Additionally, in many cases it's an interchange where one of the parties is smarter than the other one, such as this consultation you've asked me to have with you.
=== Are you one of those clods who would argue with me over the definition of "run" === ?
Hardly, but see "irony" in your dictionary, or consult the online Urban Dictionary for "self-pwn."
And don't worry Sparky, if M-W sends me an advance to formulate a new improved dictionary I'll be sure to consult you.