Why would it be incorrect to describe to the vectors of an infinite dimensional vector space as "infinite tuples"?
As far as I know, a tuple must be finite. But the objects of vector spaces are called vectors. Vectors are, generally, satisfactorily represented by tuples. In the case of an infinite vector space, however, it appears that tuples cannot be used without introducing some notion of an "infinite tuple." Why do tuples need to be finite (when a sequence can be infinite)?
Some vector spaces are represented with infinituples: in particular, sequence spaces. If you think about it, tuples from the same set can be thought of as functions from finite sets $\lbrace 1, 2, \ldots, n \rbrace$ to the set. If you want an infinite version of this, just make the set into $\mathbb{N}$. A function from $\mathbb{N}$ to a set is a sequence!
But, for many vector spaces, this infinituple idea is impossible. It all comes down to bases. In finite dimensions, we have one dominant kind of basis: the Hamel basis. A Hamel basis is a set of vectors $B$ with linear independence and spanning. Linear independence means that a null linear combination of any finite subset of $B$ must be the trivial linear combination. Spanning means that every element of the space can be expressed as a linear combination of a finite number of elements of $B$.
Note that this definition varies slightly from the one given in finite-dimensional linear algebra courses, in that it allows for infinite sets $B$, and compensates by only considering finite subsets. However, we still get a uniqueness of representation, allowing for, in a sense, coordinate vectors!
Now, it can be shown with Zorn's lemma that every vector space has a Hamel basis. Unfortunately, in infinite dimensions, it is typically an intractable problem to construct such sets, let alone do anything with their representation.
Infinite dimensional vector spaces are often considered with norms (or at least locally convex topologies), giving them a topological bent as well. Hamel bases pay no respect to topology, and consequently are rarely used. Schauder bases are more common, as they allow for infinite sums (which require convergence, and hence topology).
These infinite sums allow for countable representations, where a Hamel Basis will often become uncountable. That's another problem with these basis representations: they can easily become uncountable, and thus not really possible to represent as an infinite list of field elements. In fact, many interesting infinite dimensional spaces are Banach spaces, and infinite-dimensional Banach spaces always have uncountable Hamel bases (it's a nice exercise in the Baire Category Theorem).
Basically, to sum up:
There are prominent examples of infinite representations (e.g. Fourier series) being useful, but typically these rely on countability and the underlying topology. There is also a whole branch of functional analysis devoted to looking at various bases!