I work part time for a company that manufactures things called gobos which are glass filters that slot in front of light projectors, and these can project type fairly high quality and may be worth looking in to. You won't be limited to any kind of typeface but it can be quite expensive you'll have to either hire a projector with a gobo holder or buy a second hand one online. Let me know if you'd like any more info.

Just to elaborate (I am a theatrical lighting designer when not busy being a typophile), glass gobos specifically can project super hi-res type. You will need a theatrical lighting fixture like a SourceFour to do this.

Other than that, this has as much to do with the anti-aliasing of the program you're using for typography as with the specific font. 1920 x 1080 is full HD, so not bad at all as far as resolution goes (though type always benefits from increased resolution much more than images do). I wouldn't worry about it too much. Choose the font you want and accept that by the nature of the medium of projection, it will be made of pixels. You can also fuzz out the focus of the projector slightly to make the pixels a bit softer.

I'd assume that fonts designed specifically for onscreen reading (for eBooks, PowerPoint, etc) might be the clearest, as they are tweaked to be easily readable on monitors.

The quality of how the fonts display on an output device such as a monitor or a projector is also dependent on the quality of the font rasterizer. A font rasterizer is a piece of software sitting in the operating system of the computer which supplies the signal for the output device. Most applications (but not all of them …) rely on that built-in font rasterizer and its standard settings. So on the one hand it makes sense to look for well hinted fonts, but on the other hand it does not make sense to do so when either the font rasterizer ignores them anyway (such as current versions of Apple’s OS-X will do, because it will do a better job in most cases anyway) or is not capable to interpret them (such as older versions of Windows and Apple OS). What is the operating system and version you are working with?

When it comes to the choice of font, one may consider that although Univers is a great typeface, at smaller sizes the differences between a, e and s, as well as 3, 6, 8 , 9 maybe not clear enough because of the relatively small apertures. You will have better results with typefaces such as Frutiger, Verdana, Lucida Sans etc, because the apertures in the aforementioned characters are larger than in Univers.

The OS rasterizing process is probably not relevant in this case because I doubt you will be projecting an active text window in an actual application. (Unless you use PowerPoint etc which is a terrible idea.) You will probably want to compose your text in InDesign (make a document with the "Web" preset at 1080 x 1920 pixels), then output to a .png image at the native resolution of the projector.

Use a proper theatrical projection program like QLab (free; see figure53.com) to send this image to your projector. QLab can even be programmed to automatically start and stop the image at your gallery opening/closing times. QLab running on a Mac Mini is sort of an industry standard for things like this.