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.

The big difference between the Pico LED 1080p models and the full size projectors is going to be in brightness. For DLP, you will also get a bit cleaner of an image from the full size DLP chip.

You lose versatility compared to something like the Epson 3000 series, which is not only 3 times as bright (or more) in the real world. But, has a great lens which offers a ton of placement flexibility.

But, the little LED 1080p models are no slouches. I reviewed the PF1500U and was really impressed at the image quality which was delivered for the size and the money. It certainly does well in a dark room on a screen up to, and potentially beyond, 100" in diagonal.

Always keep in mind that projectors are designed around use in a darker space. The darker the better. Projectors with the lights on and the shades open? Well, you remember watching films in school don't you? Shades closed, lights off, and even then the quality was only so-so. That's physics.

If the brightness and placement flexibility is needed, then opt for a traditional projector. If the lower brightness can be managed with a darker space and limited size, and the placement flexibility isn't an issue, then the LED models are just a really great option. Super portable, low power, no lamps to deal with. Put it in a backpack or suitcase and it travels very well. Add a streaming stick and you have your Netflix/Amazon streaming. Hook up your phone for even more videos. Hook up a laptop for the whole Internet. Want better sound? A decent pair of computer speakers can really help out.