Or, to the opposite, was Tengwar already designed before the Voynich manuscript was rediscovered and publically known?
The manuscript was rediscovered in 1912. Tolkien, according to Wikipedia, started developing Elvish in 1910 or 1911... before the Voynich manuscript was rediscovered.
There is at least one claim that he was aware of it, though, although it would have to be after he had already started developing the languages.
This is a transcript of a message sent to the Voynich mailing list, sent November 13, 2002:
I have "lurked" on this list for a while and finally have a comment to offer, albeit a very insignificant one.
I wonder if the fact that Prof. JRR Tolkien apparently had an interest in (or at least knowledge of) the Voynich Ms has been discussed or indeed is of any interest at all?
The thing is, a nagging feeling that I had once seen the VMs many years ago - long before my recent interest was sparked by that piece in The New Scientist - is resolved. I now recall it - and though the persons concerned having died long since makes my little anecdote mere hearsay and in no way veridical, I thought I'd forward it. An old friend, a retired military man with an amateur interest in codes and cyphers, once showed me a couple of not entirely distinct b&w copies of pages from a curious coded manuscript, which I now realise were a couple of folios of the VMs. I was not especially interested in them at the time, I think, but the reason that the incident made an impression was that he said that they had been given to him by Prof JRR Tolkien. At that time I had just discovered and was very much 'into' Tolkien so I was most envious of my friend's knowing him and pressed for details of the great man, though in the end I never achieved my longed-for personal introduction. So I now wonder if there might be any reference anywhere in the mass of Tolkien papers to our VMs, and is this of any possible slight significance? After all, JRRT knew a great deal about languages and artificial scripts of course and if he was interested enough to make and pass on copies to a friend, he might have devoted some time to the VMs himself. And the Voynichese script does have a Tolkien-ish look to it or vice versa: could it have influenced him?
(this was found through a link from CipherMysteries.com)
However, since his development started before the Voynich manuscript was discovered, I'd say that it's... kinda unlikely that he was inspired by it.