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Take advantage of the large difference in thermal conductivity between copper and stainless steel (approximately $400$ and $16$ $mathrm{Wm^{-1}K^{-1}}$ respectively). If you put one end of a metal rod into contact with something held at a constant high or low temperature $T_C$, you would expect the other end to asymptotically approach that temperature like:

$$
T(t) sim T_C + A e^{-lambda t}
$$

where

$$
lambda = frac{k}{rho c_p L^2}
$$

where $k$ is the thermal heat conductivity, $rho$ is the mass density, $c_p$ is the specific heat capacity and $L$ is the length of the metal rod.

Assuming the straw is approximately $0.1$ m long, you should get $lambda$ values of approximately $0.011$ $mathrm{s^{-1}}$ for copper and approximately $0.0004$ $mathrm{s^{-1}}$ for stainless steel.

A simple experiment would consist of putting one end of the your straw into contact with a container of ice water, or maybe a pot of boiling water, while you carefully measure and record the temperature at the other end as a function of time. (If the straw is longer than $0.1$ m, immerse the extra length of the straw into the water.) The best thing would be if you had some kind of digital thermometer that allows you to log data, but it could probably also be done with an analog thermometer, a clock, and a notebook. After taking the measurements it should be relatively easy to determine weather the temperature difference decreases with a half-life of one minute or half an hour.

There are many potential error sources in a simple experiment like this, but since the difference between copper and stainless steel is more than an order of magnitude, it should be relatively easy to tell them apart despite these errors. The experiment could also be carried out for other rods that are known to be made of copper or stainless steel (or of some other metal), to validate that the experiment gives approximately the expected result for them.

Why not density? At least for a quick check and as for the title question. You are dealing with about < 8 and 9 g per cubic cm, respectively for steel and copper.

Not overly laborious and especially non destructive at all.

To measure the volume you can submerge the object in the narrow container of your kitchen. If isn't graduated you just mark the displacement. The same with a graduated one for better precision. Collect that volume of water and move to a kitchen scale. As for you can weight the object as well, the experimental part is over.

Also note that as it is a straw it could also be possible to measure the thickness of the wall. So you calculate the volume. A caliper might be uncommon at home (incidentally I have one) but it remains a possibility. Inner diameter then is measured by passing/non passing wires. This might be tedious.

Indeed a straw complicates the procedure as its volume brings things down to ten grams or so.

See also the comment of Andrew Morton below. Having a dietary scale and using his trick he did indeed identified the nature of a straw.

What about Eddy currents? If you've got a pair of strong neodymium magnets (doesn't everyone?) - move the straw in between two magnets with their poles opposing. Copper/Aluminum will have a strong interaction and you'll feel some kind of resistance or tugging (not unlike moving it through a thick fluid). Stainless steel won't produce any such effect.

Disclaimer: I've run this experiment with solid slugs/pucks of copper, aluminium, nickel, chrome-plated steel, and #316 stainless in our shop. I haven't tested with hollow cylinders like a straw.

To see if the straw is stainless steel with a copper-colored coating, you can carefully sand or file off a bit of material from the end of the straw and see whether or not it is copper-colored throughout its thickness.

To see if the straw really is copper, dip one end of it in a boiling solution of 1/4 cup white vinegar and 1 teaspoon of salt. This mixture is commonly used to remove tarnish from copper objects and will quickly make the end of the straw very bright and shiny. Stainless steel will be unaffected (i.e., not brightened) by this.

You can measure the voltage when immersing the sample as an electrode into lemon juice or salty water and use known steel or known copper as the other electrode of a galvanic cell. Better use pure copper as the other electrode as different steels have different electrochemical properties. Some uncertainty, whether you don't just have steel with the same electrode potential as copper, may remain.

Here's an idea (this is not my area however): Consider using the difference in resonant frequency. This depends on length ($L$), Young's Modulus ($E$), mass per length ($M/L$, requires mass of straw, likely found from manufacturer or else use e.g. a food scale) and second moment of inertia ($I$, area moment, unique to the geometry, a tube of known inner and outer diameter can be calculated). Copper has $E = 36times 10^6$ PSI, stainless steel (according to Engineering Toolbox) has $E = 26 times 10^6$ PSI.

For a tube of inner diameter ID and outer diameter OD, the second moment is
$$ I=frac{pi (OD^4 - ID^4)}{64} $$
Note that those are fourth powers of the inner and outer diameter.

For $E$ in PSI (pound-force per square inch), $L$ in inches ($text{in}$), $I$ in $text{in}^4$, and $M$ in $text{lbm}$ (pound-mass), for a simply supported straw (note: length is length from support to support), we can calculate the resonant frequency by (source):
$$ f = frac{pi}{2 L^2}sqrt{frac{EI}{M/L}} $$
Where $f$ is the frequency in $text{Hz}$.

The challenge might be supporting the tube properly, maybe fix the ends internally, or with some steel wire (or paperclips). The resonant frequency can (hopefully) be measured by your phone, with an audio spectrum analyzer.

For an example, consider this straw from Amazon. $OD = 0.3125text{ in}$, assume $ID = 0.3text{ in}$ for sake of example, $L = 8.5text{ in}$, and let $M = 0.01text{ lbm}$ (product net wt, not sure what packaging it has). It's made of food-grade steel (18/10, 303 grade) with $E = 27-29times 10^6text{ PSI}$. The second moment is calculated to be $I = 0.00017text{ in}^4$. Assuming the straw is ideally supported, the stainless steel straw would have a resonant frequency around $43-44text{ Hz}$, while a copper straw would have a resonant frequency of around $49text{ Hz}$, assuming I haven't made a mistake in my calculations.

Touch it with a magnet. Steel is magnetic. Copper is not.

Copper is much more malleable and ductile than stainless steel, so a very simple but possibly destructive test would be to try bending the straw with your fingers.

A copper-plated steel straw will still have the strength of steel. A solid copper straw would be much softer and easily bent.

For comparison, try bending a paperclip, then try bending a piece of copper wire the same thickness as the paperclip. The difference is very noticeable.

Taste it, copper and stainless steel taste nothing a like.

I'm surprised no one has mentioned hardness, since this is what metallurgists and machinists would use. Basically, you can buy sets of files made out of different grades of known metal hardness. If the file scuffs the metal, the metal is softer than the file. If the file doesn't scuff the metal, instead the metal smooths down the file, the metal is harder than the file. You work down from the hardest file in the set down to the softest, until you don't see any more scuffing, and bingo, that gives you the hardness of the metal, with some degree of accuracy.

Having said that, steel is typically so much harder than copper, that it should be incredibly obvious from using ANY steel file, whether you are struggling to cut through steel, or chew right through copper. Copper is so soft you can usually bite it and leave a mark, although this isn't recommended. Use a file to do the biting for you.

You might want to try soldering - If the solder drop attaches easily to the straw it's copper.

This is somewhat destructive of course, but since others pointed out, that copper is unsafe to drink from, you would want to throw it away anyway.