Can a noun (such as “duct tape”) be used as a verb?

This is, I learn, called conversion in lingustics:

also called zero derivation, is a kind of word formation; specifically, it is the creation of a word from an existing word without any change in form

making a verb out of a noun:

Verbification, or verbing, is the creation of a verb from a noun, adjective or other word.

In English, verbification typically involves simple conversion of a non-verb to a verb. The verbs to verbify and to verb are themselves products of verbification (see autological word), and—as might be guessed—the term to verb is often used more specifically, to refer only to verbification that does not involve a change in form. (Verbing in this specific sense is therefore a kind of anthimeria.)

Examples in English number in thousands and this is a very potent source of neologisms, due to the fact that newly coined words take a well defined meaning from the noun. It is typically used when there is no ambiguity (compare to tape with to glass; while tape is an object with clear dominant use, glass is not so it is not so effective).

Of course, prescriptivists oppose it on principle, but also others oppose it when it is not done 'in the spirit' of the language, for example the following quote from Bill Waterson's:

Calvin: Verbing weirds language.

is perfectly understandable, but it sounds very strange.

In cases where it does not weird it - it is easily accepted. In your example, the way the noun 'tape' was verbed into verb 'tape' provides an established path for "duct tape" to follow.

Yes, using "duct tape" as a verb is quite a common practice. In particular, you will frequently see it together with "together" as in your example, but there are other usages too. In fact I probably hear it more as a verb than a noun. Consider this example:

Hey Caleb, can you duct tape this cord to the floor so people don't trip on it?

Or more generically:

How should I do ..........?
Easy! Just duct tape the .......... to the ...........

These likely fall in the realm of colloquial language, but are quite common, easily understood, and don't strike even a picky ear as out of place.

By analogy with "glue" or "tape" or any other fastening method, sure. As Unreason notes, nouns get verbed in English a lot.

However, note that the verb form of a two-word noun takes a hyphen: "duct-tape", not "duct tape".

Tangent: "It's not the verbing that weirds the language, it's the renounification." -- Marc LeBlanc

Also, the adjectival phrase phrase "duct-taped together" is pretty commonly used as a synonym for improvised or jury-rigged.

Such constructions are not uncommon: "a million-to-one chance", "a red-and-green Christmas bow", and so on; it's more compact than "the chances are a million to one" or "the Christmas bow was red and green".