This is primarily a font problem. It generally takes more than ten years from the adoption of new characters into Unicode to their widespread availability in fonts. (Optimistically speaking. And I’m not joking.)
It’s also a browser issue, in the sense that when a web page does not specify fonts, or specifies them generically only, such as sans-serif, browsers will apply different strategies. When their default font does not contain all the characters present on a page, they will use different fallback fonts. And they may fail here—failing to render a character even when some font in the system contains it.
What you can do as a user is to download and install fonts that contain the characters. There are not many of them for Linear B; see
http://www.fileformat.info/info/unicode/char/10083/fontsupport.htm
As far as I know, the list on that page is complete except for the Alphabetum font and its free trial version Alpha-Demo, see
http://guindo.pntic.mec.es/~jmag0042/alphaeng.html
But there might be some even more rare fonts that support (some of) Linear B.
You can also change your browser’s default fonts to view a page with such characters if it does not otherwise work. This is inconvenient because special fonts containing special fonts are not that great for normal browsing.
What you can do as a web author who puts such characters on a page is to declare a suitable font list, or “font stack”. For generalities, see my page Guide to using special characters in HTML. In this case, I would recommend the following (for elements containing Linear B characters):
font-family: "Everson Mono", Code2001, "MPH 2B Damase",
Alphabetum, "ALPHA-Demo", LastResort;
Still, only a small fraction of visitors would see the characters right, so you might consider including some links to pages where suitable fonts can be downloaded.