I want to say "people continue to study Latin", and I came up with lingua Latina studenda pergit. Is such a combination gerundive and pergere grammatical? It must be understandable, but I do not recall ever seeing it used or discussed. Another option would be lingua Latina studeri pergit, but I want to know if the gerundive works, too.
I don't think so. First, are you sure that pergit can be used in this sense? I would use pergo to indicate a sense of progression, and I would have an agent in mind as its subject, someone who continues to move in a certain direction. Perhaps permanet or perstat would fit?
As to the gerundive construction, i.e. a gerundive that is used dominantly, I believe it cannot be used in the nominative, nor in the accusative without a preposition. The same applies a to gerunds, mutatis mutandis. In its non-dominant use, a gerundive has a sense of obligation or prediction, which, as you say, is not what we want in this example: "the Latin language continues to be compulsory to be studied"; or as an ablative: "it continues while studying the Latin language".
But the passivity of the gerund is also problematic here, because studeo is normally intransitive: it goes with the dative and cannot normally be passivised with a subject. A gerundive is also passive, but studendus and linga Latina could not agree in case. Then what would the gerundive agree with?
In the nominative, one would use a simple noun or an infinitive rather than a gerundive, to express "doing something" as a subject or direct object. In this case, a noun comes first to mind:
Studium lingae Latinae permanet.
An infinitive is possible in theory:
? Studere linguae Latinae permanet.
Somehow, I don't like this very much, but it's hard to explain why not. Perhaps verbs that mean "continue" are not often used with infinitive constructions? At least I don't think permanet is constructed with an infinitive; and perstat with an infinitive means something else, as does pergit.
Linguae Latinae adhuc studemus / semper studebimus.
This has a slightly different meaning, but it could be appropriate.