Over a ground field of characteristic zero, the algebra of differential operators $\mathcal{Diff}(X)$ on a smooth affine variety $X$ is generated by the functions $O_X$ and the derivations $Der(O_X)$, both viewed as subspaces of the endomorphism algebra $End(O_X)$ of the vector space $O_X$. This is a theorem of Grothendieck that one can find proved in detail in a few places, likethe book Noncommutative Noetherian Rings J. C. McConnell and J. C. Robson or, if I recall correctly, J. E. Bjork's Rings of differential operators (but maybe Bjork does only the analytic case? I don't have the book at hand)

Now there are some obvious relations one knows that hold in $\mathcal{Diff}(X)$ in that case: functions commute, the product of a function and a derivation is given by the module structure of the derivations, the commutator of two derivations is, well, their commutator, which is also a derivation, and the commutator of a derivation and a function is the value of the former on the latter.

Are these relations enough to give a presentation of the algebra?

One could phrase this in the language of Lie-Rinehart pairs: is $\mathcal{Diff}(X)$ the universal enveloping algebra of the Lie—Rinehart pair $(O_X,Der(O_X))$?

The answer to this question is yes, and it is part of what one needs to prove in order to show that the associated graded algebra of $\mathcal{Diff}(X)$ with respect to the filtration on differential order is the coordinate ring of the cotangent bundle on $X$, for example.

Has a detailed proof of this been written down?

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    $\begingroup$ Though it may involve more relations than you want, perhaps you might like EGA IV$_4$ 16.8 (see in particular 16.8.8), 16.10.1 (especially condition (ii) there), and 16.12.2. $\endgroup$ – nfdc23 Dec 11 '17 at 21:50

A paper by Vladimir Bavula addresses this question. Indeed the relations in question are sufficient to determine $Diff(X)$. Moreover, Bavula gives an explicit finite set of generators of $Diff(X)$ (functions and vector fields) and presents defining relations between the generators.

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