The Artscroll siddur indicates that the שוע on מ of למנצח (e.g. תהלים כ א) is a שוע נע, but I am not sure why this is the case. The vowel beforehand is a פתח (a תנועה קטנה) and none of the other rules I know of imply anything other than a shva nach. Related: see What are the rules for sh'va na?
According to the masoretic tradition of Tiberias, a shewa under a mem which is preceded by a word-initial הַֽ־ with a gaʿya (meteg) is often pronounced naʿ. In many of these cases, the shewa is shown explicitly as a hatef patah in the Aleppo Codex.
The word למנצח in Psalms 20 (or indeed, anywhere) does not have a ga`ya nor a hatef vowel under the lamed. Thus, there is no especial reason to pronounce the shewa as naʿ.
See: Diqduqe Hat-Teʿamim, Yeivin's Introduction to the Tiberian Masorah section 347.
EDIT: Evidently is not clear to me what is being asked. I have shown that Artscroll is wrong as far as Tiberian Massorah is concerned. If the question is indeed still why they wrote the wrong thing, the answers could be human error, misunderstanding of the Tiberian rules, or perhaps they were following a grammar that did not follow Tiberian rules here. An example of such a book is this one:
(A Biblical Hebrew Reference Grammar: Second Edition)
Here they assume that a shewa remains after degemination.
I don't think there is anything more profound here.
Rav Aharon Lopiansky, in his amazing siddur called Aliyos Eliyahu, has a section on the bottom called Mesores HaTefillah. There he explains the sources for the parts of the siddur, and variant readings.
On the word למנצח, he writes:
מ' בשווא נח עפ"י כתר ארם צובה וברוו"ה שווא נע
So it would seem then that Artscroll is going with the grammarian Rav Heidenheim instead of the earlier Aleppo Codex. He made many changes to the siddur, which many follow, even if it's a change from the past. Why he made the change is another question.
First of all, realize that there is some debate about this; the Tanach Simanim, for example, has a sheva nach on the word למנצח and the Tikkun Simanim has a sheva nach for similar examples (see, e.g., הַֽמְדַבֵּ֥ר in Genesis 45:12 but also many others). On the other hand, the Koren siuddur agrees with Artscroll.
In general, if you take a word that starts with a non-guttural letter and aשוע נע, and then add the prefix ל with a patach, then the initial letter gets a dagesh hazak and the שוע נע remains. For nouns resulting from pi'el verbs* such as מנצח and מְדַבֵּ֥ר, the initial מ does not get a dagesh hazak; there is a debate about whether the sheva remains נע (Artscroll, Koren) or becomes נח (Simanim).
*This also applies to nouns resulting from pu'al verbs.
See Gesenius 20 m and 35 b.