Both the handles of cutlery and entire spoons have been made of elephant ivory, which is from a tusk (a tooth) of an elephant. Elephant is of course not a kosher animal. Presumably, then, these spoons and cutlery would render any food eaten with them (hot) non-kosher as well (depending on circumstances). Does anyone have any sources (or additional arguments) that say as much, or that say the opposite? (Of course, for practical halacha, consult your rabbi rather than relying on what you read here.)
Bones of "neveila" (improperly slaughtered animal) that have no marrow or moisture do not impart forbidden taste (based on Shulchan Aruch 99:1) because they are not fit for eating (Taz 99:1). The Taz's reasoning should apply to the bones, tusks and other inedible parts of a temeiah (forbidden animal) as well.
I'm not sure why you seem to be looking for another answer, but here goes.
I'll start with a simple question: The Halacha is always that we follow the majority. If a court is divided, the Halacha follows the majority. If the majority of an animal's neck was slaughtered, it is kosher. If the majority of the world is not sterile, we legally assume any given individual not to be sterile. And most directly relevant to this discussion - if a non-kosher piece of meat gets mixed up with a majority of identical but kosher pieces of meat, all of them may be consumed according to the letter of the law.
This being the case, why do we say that flavor can ever render something unkosher? There is always a majority of kosher substance in the mixture!
The answer, explains the Ra'avad (quoted by the Rashba to Chullin 89b ד"ה אמר), is as follows:
The key point is that having one part nullified by the majority is only possible when the one part is not currently discernible. Since there is flavor, the whole concept falls apart.
This idea is the basis for all Halachos of transference of flavor. Anytime we say that something 'became' not kosher because it was cooked in a non-kosher pot etc. we are in essence saying that there is a small amount of non-kosher flavor that was transferred during the cooking process from the walls of the pot and into the food. It used to be that they'd have a non-Jew come and taste the food to tell if he could discern any non-kosher flavor. For a number of reasons this stopped, and as a matter of doubt we normally require 60 parts kosher to 1 part non-kosher to be able to assume that no significant amount of flavor was transferred.
[A relevant, side consideration: If you have two items which taste the same that were cooked together, while according to Biblical Law the kosher item would remain kosher since the non-kosher flavor is not truly discernible, Rabbinic Law (according to the conclusion of the Talmud as interpreted by most Rishonim) requires 60:1 - since there was real, full-fledged flavor transferred and it just isn't discernible do to another consideration - i.e. the thing receiving the flavor already tastes that way.]
Based on the above, one can correctly conclude that in cases where there certainly was no significant flavor transfer (due to the inherent lack of flavor of the non-kosher item), no amount of cooking together or 'cross-contamination' will render something not kosher. This is the basis for the Shulchan Aruch and Rema's ruling (previously cited by @YDK) that 'non-kosher' bones do not render something not kosher even when cooked together.
I believe it is quite clear that cutlery and dishes made out of elephant tusks fall into this category. There is not the slightest chance that you will taste any significant elephant flavor in your food. Therefore they pose no kashrus concerns.
This was going to be a comment on YDK's answer, until I realized there is more that is relevant to the original question than just a comment on that answer.
Yoreh De'ah 99:1 seems to tell us that bones of Isur cannot make another food Asur; in fact, they can be combined with other Heter to nullify the Isur that they are being cooked with. Wonderful; there you have your answer, right? Bones of Neveilah do not render food Asur! So when I started 99:2, I was surprised - the Halachah seems to be that bones do absorb Isur. What does that mean, then, for them being placed into Heter? Do they make it Asur? Yes, this is the original question, but 99:1 no longer answers it alone in light of 99:2.
I think there is still an answer here that can be gleaned from 99:1, however. The bones themselves are not Asurim. In fact, they can be combined with Heter to nullify Isur. However, they are not impervious to being made "Treif" themselves. Thus, one needs to be sure, like in all other Keilim, that no Isur was cooked with them alone, ie., no part of the elephant was cooked with the tusks in fashioning the ivory into a food-utensil.