All Elvish is (probably) open source (in the US)
There is a real legal history of battles over open source languages, two threads of which are relevant here. The first is that the US court system has definitively ruled that it is not a violation of a constructed computer programming language's copyright to write a new computer program using the same language and grammar. Although there have been no definitive cases regarding 'conlangs' (constructed languages), the other thread is the situation of the legal battle over Prelude to Axanar.
General background information
There was a journal of Tolkien's linguistics titled Tyalië Tyelelliéva, originally hosted on GeoCities and now gone. This journal published original works in Tolkien's languages in addition to other analysis, but evidently ran afoul of Tolkien's estate which in 1999 took the stance that Quenya and Sindarin in particular (and presumably all Tolkien's languages in general) were copyrighted.
The journal publishers sought the legal advice of the General Counsel of the National Endowment of the Arts who sent back a legal opinion. Here is the only link I could find of this opinion; much of the hard evidence of this whole situation is shrouded in the mists of lost GeoCities. The main points of the opinion were these:
Words, short phrases, names, symbols, typefaces, and variations of lettering are not subject to copyright protection by 37 C.F.R. 202.1 (1974) [that is the title of a US Congressional act]
"In no case does copyright protection for an original work of authorship extend to any idea, procedure, process, system, method of operation, concept, principle, or discovery, regardless of the form in which it is described, explained, illustrated, or embodied in such work." (17 U.S.C. 102(b)).
Tolkien's languages, by his own self-admission, derive significantly from extant or extinct world languages, which are of course not copyright-able. The fact that many proper nouns like 'Osgiliath,'Theoden', and 'Celebrian' have an origin in real, historical languages makes claims of 'originality,' which is a necessary pre-requisite of copyright, difficult to establish.
"...the fair use of a copyrighted work, including such use by reproduction in copies or phonorecords or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright." (Title 17 U.S.C. 107) The author of the legal opinion thinks that a journal with original poetry constitutes fair use.
The journal in question here evidently folded around 2001, and no court proceedings (so far as I can determine) were filed on either side, so this is just background information and not legally binding.
For more background information, a more exhausting study was made by Harvard Law Review in 2014. A longer read, this summarized existing conlang legal actions (there have been none that actually went to court), and comes to the conclusion that copyright law is ill-suited to regulation of a constructed language, although it stops sort of giving an opinion on the legality of third-party usage of a constructed language.
The case of computer programming languages
In general, the grammatical principles of a computer language are not copyrightable. If a Python 'Hello World' program looks like
there is nothing preventing me from writing a programming language and/or compiler that uses the exact same syntax to create the exact same effect. This is the decision of Computer Assocs. Int'l, Inc. v. Altai, Inc., 982 F.2d 693, 720-21 (2d Cir.1992). Following the Supreme Court decision Baker vs. Selden which "denies copyright protection to expression necessarily incidental to the idea being expressed", this case settled on the 'merger doctrine' which states that
[C]opyrighted language may be copied without infringing when there is
but a limited number of ways to express a given idea.... In the
computer context, this means that when specific instructions, even
though previously copyrighted, are the only and essential means of
accomplishing a given task, their later use by another will not amount
So if my programming language wants to use the exact syntax
there are only a limited number of way to express this idea, and I can copy a copyrighted work (such as a copyrighted Python program that uses that exact line of code) without infringing on the copyright.
By this logic, if a court would apply it to conlangs, which has not yet been done, I could copy Sindarin sentance structure, even if Sindarin has a valid copyright, without infringing; thereby allowing me to generate original works.
The case of Prelude to Axanar
There have long been Star Trek fan films, mostly (in my opinion) terrible. Evidently, Axanar was to be a fan film with a budget of over $1 million, some serious production values, support of Important Star Trek People like George Takei, and even some actors who had appeared in other Star Trek movies. Paramount, which had hitherto been relatively tolerant of fan films, sued for copyright infringement. Paramount had a pretty strong case, since characters like Garth of Izar and fictional races like Vulcans and Klingons are pretty clearly copyright-able. Eventually, the case was settled in 2017.
However, in the course of the lawsuit, Paramount asserted a claim to control over the fictional languages. Sort of. This was probably never a claim that Paramount really wanted to make, but was just involved in the legal claptrap. The Language Creation Society (LCS) filed an amicus brief stating that conlangs were not copyrightable, and the defendants (the producers of Axanar) filed a motion that also said in part that the Klingon language could not be copyrighted. The defendent's motion was accepted by the court, so the copyright issue over the constructed language was excluded; the LCS's amicus brief was then rejected by the court as not applicable.
Ultimately, this case decided nothing. Paramount had a strong case with characters and races and organizations and probably wasn't willing to risk a negative opinion on constructed languages. If anyone is going to file a lawsuit to get open use of a constructed language, it won't be about Klingon since Paramount is a lot richer than the inventor of any other conlangs. But it is relevant that the judge was willing to accept the defendant's reasoning which referenced the same Baker vs Selden which provided the precedent with computer programming languages.
There is no official court decision on whether the grammar and vocabulary of a constructed language, being utilitarian in nature, can be copyrighted. However, there is some good evidence and legal opinions that, in the US at least, such copyright laws would not apply.
Of course someone has to test this in court. Maybe it could be you? If Tolkien's estate comes knocking, you could fire up a GoFundMe and appeal to all the language nerds out there. Your Name vs Tolkien would get its own Wikipedia page and article in the Harvard Law Review...you'd be famous!