One, cleat through.

One, because I say so.

Which straw?
One hole at the apical aspect and one hole at the axial aspect.

Two holes, just to be contrary. Suppose you start with a hollow sphere and create a small hole on the surface and another small hole at the opposite end. The sphere clearly has two holes. If we squeeze the sphere to make a cylinder, we end up with the straw with two holes.

Like every good breakfast, I am going to start with a donut. That clearly has one hole.
Now I’m going to stretch it into the shape of a straw. It still has one hole.
Unless you are counting the interstitial spacing of the plastic.in which case it has trillions.

One, because @rebbel says so.

Take the piece of dried grass that the plastic straw took its name from:
It will usually have at least one node in the middle of it, that creates an obstruction between the hollow chambers.
A piece of grass with such a node, if cut at either end, therefore has two holes.
I conclude, that even if you remove that node in the middle, it still has two holes.

One.

My dictionary says one.

One could argue it has 0 holes if we define a hole as a void that intersects with a plane.

@LuckyGuy , A donut shape is different from a straw. A donut has volume and a straw does not. Even if we replaced the donut with an inner tube, it would be fundamentally different from a straw shape, because an inner tube has a separate interior and exterior (which is why you can fill it with air). Mathematically, the straw shape is considered as a cylinder. In math cylinders do not have parallel bases on the top and bottom like a soda can. The shape of an inner tube in math is called a torus.

@LostInParadise A straw is a torus. So is a coffee cup.

@LostInParadise I imagined a donut like one from Dunkin. (Not a hollow torus.)

Now let’s turn the donut into a straw – a blasphemous statement to a pastry lover.
Imagine putting your thumbs in the center hole and your fingers on the outside. Now pinch your fingers together and squeeze the donut ring a little to make its vertical cross section be two rectangles. Now let’s squeeze it some more. Now let’s really squeeze it hard so it is a few thousand of an inch thick. You have a donut straw.

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@Caravanfan , @LuckyGuy , I was thinking of the straw as being infinitely thin, like a 2 dimensional square shape that is bent to connect two of the parallel sides.

@LostInParadise Well, that wasn’t defined in the OP. A regular straw is topographically just a torus. At this point one has to mathematically define what a hole is.

Suppose we start with something shaped like a birthday cake and we cut out a portion in the center, so we get a donut shape. Now make an opening in the side of the cake that meets the path between the two previous openings. We can twist the cake so that the new opening is in a straight line with one of the other two openings. That means that we can take any two of the openings and twist the cake so that those two openings are the start and end of a straw, with the remaining hole on the side of the straw. We can also twist the cake so that none of the openings is in a straight line with any of the others. How many holes are there? One? Two? Three? It is not at all clear.

Take the human body for example.
Are the mouth and the anus two separate holes, or just one hole?

“Hole…1. An opening into or through anything. ...” – from Webster’s New Collegiate Dictionary, 1960 ed.

This is a straw poll then…kinda!

So we have either 0, 1, 2, or infinite depending on how we define holes…

There is no straw.

There is only suck.

Namaste.

@JLoon wins.

Rejoice, for the definitive answer has been given:
https://www.youtube.com/watch?v=ymF1bp-qrjU