There's no evidence to say that it is a constructed language.
The idea was investigated in detail by John Tiltman, a British army officer who specialized in cryptography. The possibility of the Voynich manuscript being a constructed language was brought to his attention by William Friedman, an expert on the manuscript. Friedman confided to Tiltman, after decades of work on it, that he thought the writing could be a form of "synthetic universal language", and so the cryptographer expanded his investigation in that direction, writing a paper on the theory.
The basic premise behind a "synthetic universal language" is to use characters to represent certain logical ideas; by chaining together letters, you can create "words" that describe more complicated principles. By adding the appropriate prefixes and suffixes, characters can be quite powerful. Tiltman noticed that some of the patterns in the words fit this type of design. However, the evidence for this sort of construction was not strong enough, and Tiltman's analysis of the frequency of the occurrence of certain letters did not support Friedman's hypothesis.
Tiltman also traced the historical origin of a "universal language", to determine if such an idea had been bounced around early enough. However, this proved difficult, and Tiltman found no concrete examples prior to the mid-17th century. This is two centuries after the Voynich manuscript was written - sometime in the 15th century - and so if the manuscript was indeed written in a universal language, it would predate these universal languages by two centuries.
It was clear that the productions of these two men [Wilkins and Dalgarno, two early proponents of universal languages] were much too systematic, and anything of the kind would have been almost instantly recognisable. My analysis seemed to me to reveal a cumbersome mixture of different kinds of substitution. When I was
attempting to trace back the idea of universal language, I came upon a printed book entitled The Universal Character by Cave Beck, London 1657 (also printed in French in the same year). Cave Beck was one of the original members of the British Royal Society and his system was certainly a cumbersome mixture.
Beck's writing claims that the idea of a universal language goes back to the 16th century, but this still places it many years after the manuscript's origin. The idea of synthetic universal languages, though perhaps relatively widespread in the 17th century, simply was unknown in the 15th. This was a strike against Friedman's theory. If the Voynich manuscript was a synthetic universal language, it would have borne no relation - in time of conception of structure - to later examples.
The universal language route is of course not the only possibility; constructed languages can take many forms. However, it was possible the most promising option, and research into it went nowhere. Friedman's claim remains unsupported, and, to my knowledge, no further academic analyses have been done. We certainly can't rule out the theory that the Voynich manuscript is written in a constructed language, but we have no reason to believe that it is.