I hate to node in response, but there are some factual misrepresentations in the other writeups here that should be cleared up.
"(Flutter-tonguing is) generally only used in modern pieces..."
The statement here is correct only if what we mean by "modern" here is within the last century and a half. The earliest example of flutter-tonguing of which I own a full score is Richard Strauss's Don Quixote, which is a symphonic poem of sorts based upon Cervantes's novel. In the second variation of this piece, Strauss uses a technique Schoenberg would later refer to as Klangfarbenmelodie, which can be translated as "tone color melody." Which is to say, Strauss creates a melody of sorts by having different instruments of different timbres play the same note (pitch) but at different times, sequentially, so that a linear connection can be heard as a single note is passed between the different timbres.
Now, there is no necessary link between Klangfarbenmelodie and flutter-tonguing, but in Don Quixote, several of the instruments participating in the second variation instance of the technique are using flutter-tonguing (which, as a side note, is notated for winds much as the string tremolo is for bowed instruments, with three diagonal slashes above or below a note, across any stem which might be present). Consistent with the previous nodes on this subject, most of those instruments are brass. But, there are also flutter-tonguing clarinets and bass clarinet.
Admittedly, I am not a scholar on historical woodwind technique, but as far as these "advanced tonguing techniques" go, they usually progress from instruments where they are easier and simpler to accomplish (i.e. brass instruments, members of the flute family) to instruments where they are not so easy or so simple. So the presence of clarinet flutter-tonguing in Don Quixote, which was composed in 1896 and 1897, suggests to me the technique was common even before that for flute and brass players. Flutter-tonguing's instances before that would probably be found in a genre similar to Strauss's -- which is to say, in the realm of the German/Austrian symphonic poem -- where a little crudeness in tone color would be acceptable if it achieved some higher, narrative purpose.
In my own playing experience, Stravinsky makes use of the technique himself in his infamous Le Sacre du Printemps (1913) for the English Horn and Oboe as well. So flutter-tonguing was an established -- though by no means frequently-used -- technique well before dixieland or traditional jazz really flourished.
"I don't think it's possible for other woodwinds besides the flute and piccolo..."
As one might have gathered by now, it not only is possible, it's nearly a standard technique. Granted, the technique on reeded instruments (clarinet family, oboe family, saxophone family) is probably more difficult than it is on other instruments. The reason for this, in part, has to do with a lack of uniformity in practice.
By this I mean, there is little consensus on how flutter-tonguing is achieved. That this is the case can be observed in the previous two writeups. The first describes flutter-tonguing as the quick flicking of the tongue up and down, which must necessarily be done at the tip of the tongue (unless you have an unusually agile and attached tongue). The second compares flutter-tonguing to the rolling of one's r's. Another method sometimes used is often described as the "uvular roll." These three methods involve placing the tongue in three different places, shaped in three different ways, with three different kinds of breath control.
What these three techniques have in common is that they all form what might effectively be called a "second" reed within the mouth. A quick physics lesson:
A reed works by blowing through an opening created by the reed and either the mouthpiece (as in the saxophone or clarinet) or another reed (as in the oboe or bassoon). When the air speed is fast enough, the air pressure within the reed drops to a point where the air pressure outside the reed forces the reed to collapse; this stops the air stream, allowing air pressure to stabilize before the reed snaps back into its original position, which it does because of silica within the cane of the reed itself (which is why arundo donax is used for reeds, as opposed to say, plastic straws). Once the reed is open again, the air stream continues and the cycle is repeated. Thus, noise is made.
In a similar way, flutter-tonguing is achieved by creating a similar space within the mouth with the tongue. The tongue blocks the air stream; the air stream pushes the tongue out of the way; the air stream lowers the air pressure, allowing the tongue to snap back into position (this time because of muscle tension); and so on.
So, with reeded instruments, the first two techniques are nearly impossible, for the reason that... well... there's a reed in your mouth! In order for one's tongue to be able to perform its duty as a "second" reed, it must be out of the way of the reed, which often means pulling the reed so far out of an appropriate place on the embouchure that playing becomes nearly impossible. But note that I say only "nearly." Either one of these first two techniques is used by most of the reeded instrument players I know, which often results in a very obnoxious tone with air escaping around an over-extended embouchure.
The final technique keeps the tongue in its normal position, does not require over-extending the embouchure, and is generally easier to execute anyway. I understand the technique parallels a European phoneme -- I believe it's the German "ch" as it occurs at the end of a word. In any case, it is the easiest technique to execute while holding a reed steady.
So it's often said that flutter-tonguing (and while we're talking about it, double-tonguing, triple-tonguing, and circular breathing) is "impossible" for reeded instruments, when in reality, it only seems that way because, for whatever reason, the simplest way to achieve the effect aurally (that is, the simplest way to make the "flutter-tonguing" sound whether you're technically flutter-tonguing or not) is not being taught to woodwind instrumentalists.
That said, it is still difficult to flutter-tongue on the Oboe or English Horn, which are my instruments, since they require a very high air speed just to work in the first place, and the act of flutter-tonguing deals a significant blow to the speed of one's air. One has to blow very, very hard.