Firstly, you have to realize that the purpose of a dictionary is to describe the word in such a way that people can understand the object associated with the word. A "sink" must contain the objects being washed or it wouldn't be a sink. The inclusion of "in" is helpful in conveying this and removing it would impact the effectiveness of the definition:
1) a large open container [...] that you use for washing dishes in
2) a large open container [...] that you use for washing dishes
The "large open" modifier implies that you put the dishes into the sink but including "in" makes it much more explicit. Otherwise, you could have a large open container that washes dishes akin to a washboard. While it wouldn't make a whole lot of sense, the dictionary is specifically trying to associate the preposition "in" with "sink" because that is how you use a sink: You put dishes in the sink in order to wash them.
For common, everyday usage you could remove "in" and no one would be confused but this pattern is common in dictionary entries.
Simply put: without the preposition, the infinitive is either a truly intransitive verb, or a truly transitive one; with the preposition, it is what is sometimes called a prepositional verb, i.e., a verb that—similar to how phrasal verbs work—must be paired with a preposition to take an object. The object is then not a direct object, but it is still semantically (and to a certain extent also syntactically) the object of the verb.
1A. I need a pen to write with.
1B. I need a pen to write.
2A. I have a book to write in.
2B. I have a book to write.
The difference here is clear: in the A sentences, it's explicitly stated that the pen is the instrument with which you will be writing and the book is the place you'll be writing. In 1B, a pen is just given as something that you need to have in order to carry out the act of writing; other things in that category would be paper, a desk, a chair, etc. (or these days, a laptop). In 2B, the book is simply the object of the verb.
In other words, when there is no preposition, you have to look at the constituents in the sentence to determine whether the verb is transitive or intransitive—if it's intransitive, the infinitive is often an infinitive of purpose (meaning in order to). When there is a preposition, you can usually assume the topic mentioned before the infinitive is part of a prepositional object, that is, the verb is acting in a mostly-transitive way, and the entire prepositional phrase is the object.
So what, then, of your example?
New York is a great place to live (in).
This doesn't seem to fit either: live in this sense does not take direct objects, and yet a place has no preposition.
This is because place isn't really and truly acting as a normal noun—it's more like the nominal form of the adverbial entity (some)where/here/there. Live in the sense we’re looking at here takes an adverbial phrase complement (like somewhere), but an adverbial phrase cannot be qualified by an adjective (well, it can; but its meaning changes a bit then), so a generic noun is substituted, acting as a stand-in.
If you replace place with a more regular noun that doesn't have this property, you'll see that they don't work:
*It's a great city to live.
*It’s a great street to live.
None of these two work, because you cannot *live a city/street the way you can live a place: a place is basically a noun phrase that has been semi-frozen as a pseudo-adverbial, acting like (some)where, so it takes no preposition. Cities and streets, though, do not do this, and they need to be part of a prepositional clause to be used adverbially like this. What does work is using an adverbial:
It's somewhere to live, I guess.
Naturally, if you use live as a transitive verb with a direct object, it works fine:
You only have one life to live!
And since place is still also a noun and can easily function as such, you can also use the prepositional verb live in [noun phrase] with it as the noun—which is why the double forms with and without the preposition are possible in your example.
Best Answer
I think the preposition is necessary. The analogy with case in point is not quite apt as follows: "In which city" can be replaced with "Where", hence "Where are you located?" is correct but "In where are you located?" and "Where are you located in?" aren't. Similarly, "What time is the meeting?" or "When is the meeting?" is correct, but not "At when is the meeting?"