This paper/article seems to contain the answer
There are links between folate and BH4 but some of that article is not correct.
The links are :-
1) Structural similarities, both being pteridine derivatives
2) The enzyme dihydrofolate reductase (DHFR) plays a role in both folate and BH4 related pathways
3) MeTHF is a peroxynitrile scavenger and so has BH4 sparing properties.
The article illustrates some of this correctly.
Fig 5 shows the structural similarities.
When BH4 acts as a cofactor in neurotransmitter synthesis (as shown in Fig 3 in the paper), it is oxidised to qBH2 (quinoid BH2).
qBH2 is regenerated to BH4 by the enzyme dihydropteridine reductase (DHPR; as shown on the left hand side of Fig 4b in the paper).
qBH2 can convert to BH2 (very closely related but not identical). BH2 can be regenerated to BH4 by the enzyme DHFR, as shown in Fig 4 b and 5.
DHFR is best known for its role in the folate cycle, where it reduces dihydrofolates (DHF) from, for example, vegetables, to tetrahydrofolate (THF), which then feeds into various folate pathways.
The broad specificity of the enzyme means it can also recognise folic acid, even though this compound is unknown in nature.
The same broad specificity means that it recognises BH2, reducing it by the same mechanism used for DHF.
So the same enzyme is used in two, independent metabolic pathways, a not uncommon situation.
Reference 37 in the article is about the peroxynitrile scavenging properites of MeTHF but the article seems to miss this completely and claims it as something else (more below).
As well as this midunderstanding, here are the things that are misleading in the article.
It (and Yasko) claims that there is another link between the pathways through MTHFR and that MeTHF regenerates BH4. This is shown in Fig 4a, the right hand side of Fig 4b and the left-hand side of Fig 6.
This is the idea that MTHFR runs backwards, directly regenerating BH4 (the reaction supposedly adversely affected by the SNP A1298C).
(Yasko says the reaction converts BH2 to BH4. The paper says it converts qBH2 to BH4).
Way back, I read a number of original papers trying to get to the bottom of this, since it contradicted textbooks and every scientific study I looked at. There was one exception.
That was an
in vitro study which used qBH2 and various other reactants under conditions which forced the enzyme to run backwards. (The author was interested in the mechanism of action of the reaction). qBH2 was converted to BH4.
So the claim is not entirely baseless (though the SNP claim seems to be - I could find nothing whatsoever about that). Since 1980, however, when that single study was done, no other studies have followed so it doesn't appear to be of any significance.
The enzyme has never been shown to run backwards
in vivo and the diagrams in the article are really fanciful.
The article also talks about the link between folate and BH4 via NOS, saying
It has been demonstrated that folate, in the
form of 5-MTHF, regenerates oxidized BH4,37 and in
the absence of an adequate amount of BH4, 5-MTHF
“stands in” for BH4 at the enzyme level.38
I read both those references. The first one (37) doesn't say what the article says it does (and illustrates later with another version of MTHFR running backwards). The study actually shows that MeTHF protects BH4 in the NOS reaction by scavenging peroxynitrile. This is something which others studies have shown also.
As for the substitution claim which you are sceptical about
@MacGyver, reference 38 provides
in vitro and theoretical (computer modelling of the two substances in the enzymic binding site) evidence to support this, though none that it happens
in vivo. So it is possible, but needs more study to confirm.