Maybe it is an interference from my native tongue, but why is it correct to say “It is the locations that make this tournament special”. Since there is the pronoun “it” for singular, and “they” for plural, why don’t we use “they” with a plural noun?
There is nothing wrong with the sentence. The subject is locations, which is plural and which is in accord with the verb.
The use of it as the first word in the sentence is a red herring. This is an example of a cleft sentence; it is not the subject.
It is the locations that make the tournament special.
→ The locations [are what] make the tournament special.
It is the location that makes the tournament special.
→ The location [is what] makes the tournament special.
In English, a cleft sentence can be constructed as follows:
it + conjugated form of to be + X + subordinate clause
where it is a cleft pronoun and X is usually a noun phrase (although it can also be a prepositional phrase, and in some cases an adjectival or adverbial phrase). The focus is on X, or else on the subordinate clause or some element of it. For example:
It's Joey (whom) we're looking for.
It's money that I love.
It was from John that she heard the news.
Furthermore, one might also describe a cleft sentence as inverted. That is to say, it has its dependent clause in front of the main clause. So, rather than: Example:
We didn't meet her until we arrived at the hotel.
the cleft would be:
It wasn't until we arrived at the hotel that (or when) we met her.
That sentence is actually not correct, but it isn't because of the choice of "it".
The reason "it" is used in that context instead of "they" is because the sentence is actually the answer to the (implied) question:
what is it (that makes this tournament special)?
This is one of those idiomatic things in English. This general question ("what is it that ...?") is pretty much always phrased in the singular*, even if the speaker is well aware that the answer might be multiple things. The result is that the answer is always going to be phrased in the singular ("It is ... that ...").
The reason your original sentence is not correct, however, is (as you have noticed) the subject and object do not agree ("it" and "locations"). Since this sort of sentence must always be constructed in the singular ("It is ..."), the following noun must also be phrased in the singular to be correct. There are a couple of ways to do this. You could say:
It is the location which makes this tournament special
This can, in many cases, be used even when there are multiple locations involved, and it will still make sense to the listener. On the other hand, if one wishes to emphasize that there are multiple locations, another way to say this would be something like:
It is the combination of locations which makes this tournament special
It is the choice of locations which makes this tournament special
Basically, you would use some grouping phrase to change the plural collection of things into a singular group noun so that it can be used in this sort of sentence.
[*] Technically there are some cases where this sort of question is phrased in a plural way, but it's generally only if somebody is asking for a specific number of answers (e.g. "What three things are ...?"). In this case, one will generally answer by explicitly re-iterating the whole question (i.e. "The three things which make this tournament special are location, location, location!"). You still wouldn't ever say "They are ... that ...", even when responding to a plural question.