Is there any benefit in learning Pali or Sanskrit?

I agree with other answers which state that knowing some terminology and vocabulary is useful, but learning the grammar is unnecessary, unless you're a scholar. The reason is that some words are technical terms with specific meanings, that can't be easily translated to plain English words.

While there's no need to know the grammar, it may be useful to be able to separate words from their inflection (e.g. noun declension and verb conjugation), so that you can refer to them in a dictionary or glossary.

Here's an example:

From AN 10.58 (translated by Bhikkhu Sujato):

Reverends, all things are rooted in desire.
chandamūlakā, āvuso, sabbe dhammā

Initially, when reading only the English translation, you might wonder what "things" are, and what "desire" is. Probably, this is related to the second noble truth, which states that the cause of suffering is craving.

But if you look at the Pali words here, we find that "thing" refers to "dhamma" and "desire" refers to "chanda".

The word "dhamma" has many meanings as can be seen in this answer.

The word "chanda" for desire, is different from the word "tanha" for craving (of the second noble truth). Please also see "Difference between desire (chanda) and craving (tanha)?". These are technical terms.

So, this phrase doesn't refer to craving.

But it was not clear to me what "things" refer to, so I created this question.

By the way, the "ā" in "dhammā" for "dhamma" (thing), is the declension to indicate that it's plural i.e. "things", but we can leave that detail to the translator, though it's useful to extract the basic word so that we can refer to it in a dictionary or glossary.

If the question is for academic research purpose, I will add one advantage in learning Sanskrit - it will greatly aid in researching ancient Buddhist documents in Nepal, where Sanskrit continues to be the liturgical language of Newar Buddhists.

No, I would say this is a trap, a side-trail that does not lead to the goal.

It looks attractive to the ego, because it stimulates certain hope in us, a hope to get closer to the Truth -- but this is a false hope. There is a Pali term, papañca, which can be translated as "conceptual bullshit". This is the danger here, getting caught up in quasi-intellectual masturbation.

The Truth is known by realized teachers, this is whom we must seek. Teachers show us how theoretical dharma maps to real life. The only reason to prefer the books over teachers' interpretations, is if you don't trust the teachers. If this is the case, then you should take a good look inside yourself and ask: "why do I assume that no one really understands Dharma? Could it be a projection of my distrust/aversion to this world with its people and values?" Surrendering yourself to a teacher helps.

That said, it is tremendously useful to know the main terminology. So I keep my Sanskrit and Pali dictionary links on my Google Chrome toolbar, and use them all the time. You can learn a lot from understanding the raw meaning, and the story behind, some words.

There are people who were able to turn the trap into a blessing. Take Ven. Bhikkhu Bodhi for example (the famous translator of Nikayas into English) - he got caught up in a fly trap, but because he was so determined, he was able to tear through all words, understand the meaning, and get to the nectar -- while benefiting all of us. But unless you are a super-geek like him (I respectfully bow at his feet!) -- your time is much better spent putting dharma to practice.

I agree with those who praise learning Canonical languages. Studying Pāli texts in Pāli is very rewarding. I have found my understanding of the Buddha's teaching blossomed through reading Pāli texts. I enjoying reading texts in Pāḷi and I teach beginners Pāḷi around my local Buddhist Centre.

Pāḷi is not a difficult language to learn. I would start there. It is always advantageous to learn Sanskrit because word morphology and derivations are much clearer and regular in Sanskrit. One can do without it to some extent, but especially once you start reading poetry Sanskrit comes in very useful.

What else you learn will depend on your interests. If your interest is primarily early Buddhism then the language to learn is Chinese. Not Mandarin but Middle or Classical Chinese. Studying the parallels between Pāḷi, Gāndhārī and Chinese is a hot topic at present. Often a Chinese text cannot be understood without a Pāli or Sanskrit text to compare it to. The Chinese script is difficult, but the grammar is very simply and easy compared to Sanskrit!

Sanskrit requires a much greater commitment and perseverance to learn. It is mostly written in Devanāgarī and has many more declensions and conjugations as well as a dual in addition to singular and plural. However, if you are interested in Mahāyāna texts a combination of Sanskrit and Chinese unlocks the Canon, so you cannot do without it. Sanskrit is also useful for historical interests - it is more closely related to ancient Iranian for example. Sanskrit also helps with the Brahmanical texts which were being written and discussed at various times by Buddhists. Being at least a little bit familiar with the early Upaniṣads helps to clarify some of the Buddha's teachings.

I've not tried Tibetan or Japanese. The former can be very useful for Mahāyāna texts and of course for Tibetan Buddhism. The latter has a huge body of scholarship that does not get translated into European languages. Many of the early Japanese Buddhists, like Kūkai, wrote in Classical Chinese.

If you want to know how people think, learn their language.

Learning Pali is indeed beneficial if you have a inclination to be a scholar or to teach. If it is your salvation you are after you are better off spending time practicing. But to teach others the bar is much higher than practicing on your own. This is also a worthy cause.