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I am trying to get the .bib info from a DOI, following the guide here.

Basically, I wish to run

curl -LH "Accept: application/x-bibtex" http://dx.doi.org/10.1016/j.algal.2015.04.001

to define an "Accept" header with content "application/x-bibtex".

With Mathematica. I try

 URLRead[HTTPRequest[URL["https://doi.org/10.1007/s00209-015-1612-7"],
 <| "Method" -> "GET",
 "Headers" -> "Accept: application/x-bibtex",
 "ContentType" -> "application/x-bibtex"|>]]

but this fails, with error message

"Message text not found -- (Accept: application/x-bibtex"

The output I expect is a string, with the contents:

@article{Kouhia_2015,
 doi = 10.1016/j.algal.2015.04.001,
 url = http://dx.doi.org/10.1016/j.algal.2015.04.001,
 year = 2015,
 month = jul,
 publisher = Elsevier BV,
 volume = 10,
 pages = 41--47,
 author = Mikko Kouhia and Henrik Holmberg and Pekka Ahtila,
 title = Microalgae-utilizing biorefinery concept for pulp and paper industry: Converting secondary streams into value-added products,
 journal = Algal Research

EDIT: I also want this to work with other journals. curl can handle that as well, as there seem to be a standard to request this particular format.

curl -LH "Accept: application/x-bibtex" https://doi.org/10.1090/s0002-9947-96-01558-9

It seems you are trying to profit from DOI Content Negotiation described here.

Content negotiation allows a user to request a particular
representation of a web resource. DOI resolvers use content
negotiation to provide different representations of metadata
associated with DOIs.

A content negotiated request to a DOI resolver is much like a standard
HTTP request, except server-driven negotiation will take place based
on the list of acceptable content types a client provides.

Making a content negotiated request requires the use of a HTTP header, "Accept".

After @Kuba's comment this works

With[url = URL["https://doi.org/10.1007/s00209-015-1612-7"],
 URLRead[
 HTTPRequest[
 url
 , <|
 "Method" -> "GET",
 "Headers" -> "Accept" -> "application/x-bibtex"
 |>
 ]
 , "Body"
 ]
 ]

Other content formats defined by crosscite.org are:

 "RDF XML" -> "application/rdf+xml",
 "RDF Turtle" -> "text/turtle", 
 "Citeproc JSON " -> "application/vnd.citationstyles.csl+json", 
 "Schema.org in JSON-LD " -> "application/vnd.schemaorg.ld+json", 
 "Formatted text citation " -> "text/x-bibliography", 
 "RIS " -> "application/x-research-info-systems", 
 "BibTeX " -> "application/x-bibtex", 
 "CrossRef Unixref XML " -> "application/vnd.crossref.unixref+xml", 
 "DataCite XML " -> "application/vnd.datacite.datacite+xml", 
 "ONIX for DOI " -> "application/vnd.medra.onixdoi+xml"

With[url = URL["https://doi.org/10.1007/s00209-015-1612-7"],
 Import[
 Echo@StringJoin[
 "https:",
 First@Select[
 ImportString[
 URLRead[HTTPRequest[url], "Body"]
 , "HTML", "Hyperlinks"
 ], StringContainsQ[#, "format=bibtex"] &, 1]]
 , "String"
 ]
 ]

This obviously will only work when the DOI link redirects to a journal website with a particular link structure. Notice that the BibTeX content is different.

request = 
 HTTPRequest[
 URL["https://doi.org/10.1007/s00209-015-1612-7"], <|
 "Method" -> "GET", 
 "Headers" -> "Accept" -> "application/x-bibtex", 
 "ContentType" -> "application/x-bibtex"|>];

URLRead[request, "Body"]

(*
 @articleBlasiak_2016,
 doi = 10.1007/s00209-015-1612-7,
 url = https://doi.org/10.1007%2Fs00209-015-1612-7,
 year = 2016,
 month = jan,
 publisher = Springer Science and Business Media LLC,
 volume = 283,
 number = 1-2,
 pages = 601--628,
 author = Jonah Blasiak,
 title = Haglund's conjecture on 3-column Macdonald polynomials,
 journal = Mathematische Zeitschrift
 
*)

request = 
 HTTPRequest[
 URL["https://doi.org/10.1090/s0002-9947-96-01558-9"], <|
 "Method" -> "GET", 
 "Headers" -> "Accept" -> "application/x-bibtex", 
 "ContentType" -> "application/x-bibtex"|>];

URLRead[request, "Body"]
(*
 @articleFomin_1996,
 doi = 10.1090/s0002-9947-96-01558-9,
 url = https://doi.org/10.1090%2Fs0002-9947-96-01558-9,
 year = 1996,
 month = sep,
 publisher = American Mathematical Society (AMS),
 volume = 348,
 number = 09,
 pages = 3591--3621,
 author = Sergey Fomin and Anatol N. Kirillov,
 journal = Transactions of the American Mathematical Society
 
* )