We cannot go into all of the factors that require something to have tevillat keilim, but we will give some guidelines, especially on the issue of direct contact, to which you refer. First, we should point out that only keilim (utensils) made of metal or glass need tevilla (immersion). An additional issue, which is important but beyond our present scope, is that disposable items, even if they temporarily have the same function as standard utensils, are not categorized as keilim and do not require tevilla.
The mitzva of tevillat keilim is found in the Torah in the context of the mitzva of hechsher keilim, removing through heat any taste of a non-kosher food that might be absorbed in the walls of utensils.[1] Rabba bar Avuha states that tevillat keilim applies even to new pots,[2] implying that immersion is necessary regardless of whether anything non-kosher is in the walls. Rav Sheshet asks that if that is the case, then perhaps utensils of all sorts should require tevilla. The answer given is that the Torah was referring only to klei se’uda (literally, utensils of a meal).
Understanding the reason why tevillat keilim applies specifically to klei se’uda may help us define this category of utensil. Rashi[3] says that since the Torah describes utensils that are exposed to a flame, it must be talking about utensils that are involved in a meal (apparently, including its preparation). The Pri Chadash[4] asks that there are utensils that are used in connection with heat but are unrelated to food. Therefore, he prefers the Rashba’s explanation, that these p’sukim are dealing with the type of utensil which belongs to the category of those things that may require kashering, namely, utensils for food preparation and eating.
Taking a very narrow view of the Rashba, one might conclude that the kli[5] must come into actual, physical contact with food. However, poskim understand that we are talking about the category of a utensil, namely, one which is used directly in connection with food, whether or not there is actual, physical contact.
This understanding of klei se’uda contains elements of both leniency and stringency. Does a can opener require tevilla? On one hand, if it is being used to open a can of tuna fish, the can opener almost always touches the product. On the other hand, its function is, by design, not connected to the food but to the can. So presumably, incidental contact with food should not make the can opener need to have tevilla. Indeed, the standard p’sak is that it does not require tevilla.[6] If, conversely, one covers a baking pan with waxed paper, the food is nevertheless considered to be baking in the pan, despite the fact that it touches only the paper, and the pan requires tevilla.[7]
This is not to say that direct contact between the food and the utensil is not an important factor in determining the requirement of tevilla; it just depends on the nature and the extent of the direct contact. For example, if a pot made of a substance that does not require tevilla is coated in order to improve its function (not just for beautification) with a substance that does require tevilla, then the location of the coating could make a difference. The Shulchan Aruch[8] implies that only when the obligated coating is on the pot’s interior is tevilla required (the Rama may argue; see commentators). Certainly, if the utensil in question is separated from the food by another kli, then it does not need tevilla. Thus, the Shulchan Aruch[9] says that a tri-pod that holds a pot over the flame is exempt from tevilla, and the same is true for modern devices for suspending pots.