Should the Necroscope books be read in the order published?

I would say yes.  I read them back when they were published, but my recollection is that they are fairly serialized with recurring characters and situations that build upon previous books.

Oh, no. No, they shouldn't. Each page, over the complete series should be read totally at random. Hope this helped.

The suggested reading order found at Brian Lumley's website is pretty much the published order (http://www.brianlumley.net/FAQ.html scroll down to #5):

THE NECROSCOPE® NOVELS:

01. Necroscope®

02. Necroscope II: Vamphyri! (spelled Wamphyri! in UK)

03. Necroscope III: The Source

04. Necroscope IV: Deadspeak

05. Necroscope V: Deadspawn

06. Vampire World I: Blood Brothers

07. Vampire World II: The Last Aerie

08. Vampire World III: Bloodwars

09. The Lost Years Vol. I (UK) ... Necroscope: The Lost Years (US)

10. The Lost Years Vol. II (UK) ... Necroscope: Resurgence Lost Years Vol. 2 (US)

11. E-Branch: Invaders (UK) ... Necroscope: Invaders (US)

12. Necroscope: Defilers

13. Necroscope: Avengers

14. Necroscope: The Touch

The NECROSCOPE® COLLECTIONS:

Harry Keogh: Necroscope & Other Weird Heroes. Contents: [...] Harry Keogh: Necroscope: Dead Eddy, Dinosaur Dreams, Resurrection. 

Necroscope: Harry & The Pirates. Contents: For the Dead Travel Slowly, Harry and the Pirates, Old Man With A Blade.

Then:

Necroscope: The Plague-Bearer

Necroscope: The Möbius Murders

----------

If you compare this with the published order you'll see that the only outlier is the collection "Harry Keogh: Necroscope & Other Weird Heroes!" which contains three Harry Keogh stories (as well as five others). But since these are side stories or fill in narrative gaps, I guess it doesn't matter to read them out of publishing order. I wouldn't lose too much sleep over that.

There's an additional story that's not yet listed on Lumley suggested reading order, "In Dublin's Fair City", which was first published in a magazine in 2017 and was then included in the collection "Necroscope: The Last of the Lost Years, Vol. I".

This is one of two collections (the other being "Vol. II") which have the same stories Harry Keogh stories that are also in "Necroscope & Other Weird Heroes!" and "Harry & the Pirates" as well as "The Plague-Bearer " and "The Möbius Murders".

However, the stories are in a different order than in the above-mentioned collections.