"Please confirm receipt of this email" or "Please confirm the receipt of this email"?

The version without the definite article is perfectly fine. But the version with the definite article is not exactly wrong, either; it is still seen sometimes, and it seems it actually used to be preferred in centuries past. But these days, the version without the definite article seems to be the more common one.

Some grammar

In order for a noun phrase (NP) to be (at least sometimes) acceptable without any determiner¹, its head noun must be either countable and in the plural, or uncountable. As CGEL says (p. 355),

^{1Determiners include (1) the possessives such as my and Alice's, and (2) the determinatives, which include the articles a(n) and the as well as words such as some, this, that, each, and every.}

NPs containing a determiner we call determined NPs. Under certain circumstances, nominals can themselves form NPs in the absence of a determiner, and we speak here of bare NPs. Nominals headed by plural count nouns or by non-count nouns can freely be admitted as indefinite by default, forming bare indefinite NPs such as new cars in We used to buy new cars, or Danish cheese in We used to prefer Danish cheese.

According to Cambridge Dictionary (the only one I know of that clearly says whether a noun is countable or uncountable in any given meaning), the word receipt has two meanings, one countable and one uncountable (see here, noting the notation '[C]' and '[U]' before the definitions).

In the countable sense, it means 'something such as a piece of paper or message proving that money, goods, or information have been received'. Clearly, that's not the the relevant meaning.

The uncountable meaning is this, and that's the one relevant for your sentences: 'the act or state of receiving money or goods'. The dictionary provides the following sample sentences: goods will be delivered on receipt of payment (= after the money is received); you have been in receipt of unemployment benefit for two months. Note the lack of determiner in front of receipt in these two sentences. The corresponding NP is therefore interpreted as indefinite.

Therefore, the question becomes, is receipt in your sentences supposed to be understood as indefinite or definite?

Indefinite or definite?

One thing to attempt would be to replace receipt by a related word that fits the context, but which is countable, and see if we would want to put an indefinite article in front of it. Unfortunately, all the alternatives to receipt I have been able to think of are either themselves uncountable (e.g. acceptance), or can be used in as both countable and uncountable nouns (e.g. delivery, acquisition). Worse, even if we were able to find an exclusively countable alternative, this method is not foolproof, as even similar words can function differently in sentences.

Unfortunately, as John Lawler said in a comment to this question,

Articles (and prepositions, and conjunctions, and other little words) are the gears in the machinery of grammar, and there are dozens (maybe hundreds) of special, idiomatic, non-logical, arbitrary rules, each covering a special context. For instance, the use of articles before disease names is quite idiosyncratic, but typical of the kinds of rules that native speakers have learned by heart as children.

Thus we shouldn't be surprised if we can't reason our way through deciding whether receipt in your sentences gets a the or not.

Usage in published literature

The only reliable way to reach any conclusions is to look at patterns of usage among native speakers. Here google books is of great help. Searching for "receipt of this", we find that the receipt of this was quite common in past centuries:

If, upon the receipt of this dispatch, … (source, 1820)
On the receipt of this information, … (source, 1824)
Mr. Coleman did acknowledge the receipt of this letter from La Chaise (source, 1810)

It is still found occasionally, as in this book from 2013:

In that event the goods must not be sold, but must be retained by the warehouseman subject to the terms of the receipt of this Article. (source)

However, the usage without the definite article is much more common in modern sources. Some examples:

Receipt of this signal causes the controller to perform an external interrupt sequence. (source), 1978)
The contractor shall respond within 30 days after receipt of this notice (source), 1984)
Table 14 displays recipients of unemployment compensation by duration of the work period and by the coincidence of work and receipt of this income source. (source), 1985)
… but no later than 60 days after the date of receipt of this Order by the President. (source), 1992)
…within 45 days from receipt of this letter (source), 2002)
If the complainant fails to complete the complaint within 30 days of receipt of this notice, … (source), 2003)

"Please confirm the receipt of this email." is indeed the more natural way,

"Please confirm receipt of this email." is like those abridged instructions they put inside buildings like "Pull door".