What does the "keep the fruit" term mean in Casino Royale?

It's not a secret message, just a simple instruction. You just need to look at the whole interchange:

Bond: Dry Martini.
Bartender: Oui, monsieur.
Bond: Wait... three measures of Gordon's; one of vodka; half a measure of Kina Lillet. Shake it over ice, and add a thin slice of lemon peel.
Bartender: Yes, sir.
Tomelli: You know, I'll have one of those.
Infante: So will I.
Bartender: Certainly.
Felix Leiter: My friend, bring me one as well, keep the fruit.

Bond has ordered a dry martini, then effectively changed his order to a Vesper. This grabs the attention of the other men at the table who order it too. Felix Leiter asks for one, but instructs the bartender to "keep the fruit", i.e. serve me the same drink, but without the lemon peel.

Edit:

For any non-English speakers in particular, the comment below from Brian S should be useful:

"Keep the X" is a common phrase in English when ordering food/drink used to mean "don't include X."

I don't think that had any kind of deeper message to it. Bond was ordering a rather complex drink from the waiter, part of which consisted of some kind of fruit (don't know what exactly, a piece of lemon peel maybe?). After that everybody else joined him and ordered the same, only that Felix didn't want the "fruit" in it.

I don't really think there was any more to it than that.

I would assume it means no lemon

As other posts mentioned, hold the fruit means without the lemon peel.

Since Bond's drink ends up being poisoned in the next scene, it could be that Felix feared that Bond might be poisoned and wanted to make sure he didn't accidentally drink from the glass intended for Bond (they were sitting right next to each other). Obviously Felix wasn't involved in the poisoning, but it does show that he's more cautious to those sorts of dangers than Bond is.

Alternatively, it could have just been a point of clarification for the viewer: having Bond have a visibly different drink than the person next to him makes it more obvious that the poison was intended for him when they zoom into the glass as he leaves the table.

there is a [slightly] deeper meaning, by using the word 'fruit' instead of 'lemon' Felix is conveying that he thinks the lemon is an excessive flamboyant element, and by inference that Bond is also excessive and flamboyant.

Felix on the other hand, is no-nonsense and straight forward, he orders a simple drink, he doesn't even bother choosing one, he's saying to the bartender, your pouring alcohol, pour me one, in simple phrases: My friend, bring me one as well, keep the fruit.

However i'm not sure 'keep the x' is that common as a phrase, at least not that I've heard, except in 'keep the change'

It's a phrase similar to, and in this case having the same meaning as, the phrase 'hold the x' which is an American English device for asking for something to be left out, or held back, especially from a food order.

The verbs 'keep' and 'hold' here are synonymous in the sense of 'have or retain possession of' and so can be used interchangeably to mean the same thing, although I'd argue that the phrase 'hold the x' as in 'hold the mayo' is more immediately understood than 'keep the mayo' through it's familiarity and use.

It may be worth adding that in Britain, at least in the north, beyond the reaches of cosmopolitan London, it's considered an affectation or at least irony to order food without one of its listed ingredients this way. But it probably wasn't.

Last word from 'Airplane': http://www.youtube.com/watch?v=TVdvyWK6NiI

At the request of the OP I'm adding this answer based on what my interpretation of this phrase in the movie meant. Bare in mind I do not drink and am quite unfamiliar with alcoholic beverages, but I do have friends who drink and my answer is based mainly on things I've picked up from listening to them talk about types of drinks they prefer.

One of the key rules to alcohol according to my friends seems to be that if you get fruit in your drink you open yourself up to relentless teasing and hazing for being a girl and/or a sissy. The only real source I have to back up this cultural view is from the TV show Scrubs where the main character J.D.'s favorite drink is the Appletini. The joke being that he naively considers this to be a "straight" drink and wonders when gays started drinking them.

Putting those two pieces of information together led me to the the jerk response that "keep the fruit" was him saying to make sure the drink was manly and contained only hard liquor, and no girly fruit.