Keep in mind the speakers in projectors generally suck, but beyond that, they are incredibly fun.
A projector is most often similar to a computer monitor. The ability to support a fire stick is not really an issue with any display. The fire stick has a standard HDMI connection which is compatible with all HDMI devices on the face of the earth. The only exception would be displays which have no speakers built in at all... No audio.
But, many projectors have speakers built in and will work fine.
Depending on budget and the size you are trying to achieve and the viewing conditions...
The Epson 2040 is a reasonably inexpensive model with cheap replacement lamps. It is bright, and well reviewed as a high value model.http://www.projectorcentral.com/Epson-PowerLite_Home_Cinema_2040.htm
The Epson 3700 is far more versatile in placement flexibility and offers stereo speakers of higher quality and is a brighter model overall...http://www.projectorcentral.com/Epson-PowerLite_Home_Cinema_3700.htm
It's a fair bit more expensive, but the lamps are similarly priced which is nice. It is a real go-to model for price/performance/versatility and would probably be my first choice.
Both of these models can hit a 120"+ diagonal screen size and will just 'work', which is cool.
A couple other options might include the LG PF1500U which has integrated 'smart' features to provide Netflix streaming internally. You can still use the Fire Stick with it if you want to. It is small, very portable, has internal speakers, and a LED lamp source which will last a rated 20,000 hours or so. Brightness is the real trade off, as well as a hit on contrast and overall image quality compared to the 3700 linked above. Mostly this is because of the smaller Pico class DLP chip used in this model.http://www.projectorcentral.com/LG-PF1500.htm
Another option, similar to the LG is the Xgimi H1.https://www.xgimi.com/h1.html
Which is not well known, but is gaining some traction as a decent 1080p model.
- Theater, whole house audio, and technology consultation during the build and installation process in the Washington, DC metropolitan area.