I know there's a whole thing about a Jew being allowed to use the kli of a non-Jew without tevilah, but I don't know the extent of that allowance. I was part of a conversation on facebook where someone said that a Jew can use a non-Jew's knife and cutting board to prepare kosher food without a problem, but I was under the impression that knives are in a special category of "DO NOT DO THIS." Anyone know what the source is for either a yes or no?
The two issues, as I see it, are -- dipping in the mikvah, and any transfer of non-kosher taste.
As for the mikvah -- if it belongs to a non-Jew and the Jew is just borrowing it, there's no obligation to dunk it. The obligation is only on Jewish-owned vessels. If the lender doesn't mind, I suppose you could borrow it and kasher it, which would take care of the kosher concerns.
Now as for if that knife isn't kosher: firstly, the knife may not be quite clean. Traditionally there were many valid concerns that knives retained grease, especially around the serrations. (And especially before stainless steel and modern detergents!) This is why the old-fashioned recommendation for kashering a knife was "first stick it into frozen earth, which will scrape off all the gunk. Then kasher it." (Or today's recommendation -- use steel wool, then kasher.) I've heard differing opinions today how much of a concern this continues to be. (And if the cutting board has lots of little grooves ... good luck getting it clean well.)
Secondly, even if I am sure the non-kosher knife is 100% clean, there is a rule that using it to cut sharp-flavored things like fenugreek, onions, or lemons will still get some of the non-kosher taste transferred from the metal into the food.
Lastly, even when dealing with cold, clean, non-kosher utensils, the rule is to use them "only occasionally"; if you use them regularly, sooner or later someone is going to use something hot (which transfers more taste). Rabbi Hershel Schachter's opinion is that once a month is the maximum that can be called "occasionally."
So theoretically if I can be certain this knife is clean, I can use it to cut a cucumber -- but not an onion! -- once a month. There are enough conditions here that I can see why many would just say "not recommended."
(See Shach, Yoreh Deah 96.21)