OP: Do you have or tried QOS (Quality Of Service) settings? The idea is to give certain data (in this case video streams) priority, above all other data, so that the video stream can maintain high quality regardless of any other data moving through your home network / LAN.
Yes, if the hopper and your Fire TV are on the same home network, the Stream stays within your LAN and never goes out to the internet. of course if you use Dish anywhere from a remote location outside of your LAN then obviously that stream has to go through your LAN out to the internet and to your mobile device or Fire TV in the remote location. And I can tell you that when it does go out through the internet, it is very good HD picture quality but other people have no clue that it's being sent through Dish anywhere to the Fire TV many miles away.
On the on the other hand, let me give you the benefit of learning this stuff the hard way: if your router is an older model, you could encounter disappointing results. I faced a similar situation using older routers then new inexpensive routers before finally deciding the spend the money for a really good router (partly because we were going to have a new live in family member use and further crowd our home network for their computer/internet needs.) The high-end router absolutely solve all of our problems, most significantly fairly poor and unreliable picture quality from both (the old now defunct) TiVo Fire TV app and Dish Anywhere.
If you do have an older router, you're going to be much better off with the modern, more powerful, better and greater WiFi covrrage routers with far more processing power to handle all the traffic and more memory for reliable, high quality video streams. The modern routers also have better user interface that make it far easier to implement built-in or your own custom QOS. Of course, a wired Network gives you the best performance, but if you must use Wi-Fi you're better off with the modern, far more capable routers on the market today that provide Superior Wi-Fi reliability and coverage, and by that I mean more or better antennas maintaining more powerful, or less crowded radio frequencies, at higher speeds at greater distances than the older routers, and it's worth the expense, especially if you're relying on Wi-Fi. 2.5GHz is far more crowded and, thefore, lower performing than 5GHz, which is much less crowded, and could provide far better performance.
My picture quality for Dish anywhere on my Fire TV is high definition and very close to the picture quality coming from the DVR or Joey to my eye. However everyone else can't tell any difference when they're watching Dish Anywhere Through the Fire TV. they don't have a critical eye, but they think the picture is just great. I will say that the Dish anywhere on Fire TV is high definition but to my eye it looks at a lower frame rate, something nobody else who has seen the picture notices.
Also you might want to try adjusting the settings on the TV input you're using for your Fire TV, or if you're using an AVR adjust the settings there through the appropriate input or scene-setting. Every device whether a Fire TV, Blu-ray player, Dish Hopper box, TiVo DVR, will all have some different settings in regards to sharpness the color and so-called tint and even brightness and contrast. I'll often be unsatisfied with a play back or streaming devices picture quality until I go in and adjust the settings then I find Excellent to very good picture quality experience.
And FWIW, I generally have more consistent and better picture quality experience because in my setups I'm running everything through either external switch video processors or AVR QDEO processors, and it's shocking how much they clean up for me. I've run video with processing off or directly to television for OTA and those subchannels are absolutely awful without processing viewing directly from the OTA antenna to the television, but they look pretty darn good going through the processors, then to the television. I guess my point is that there are a lot of factors involved in getting a good quality of picture.
Considering that Dish Anywhere uses Slingbox technology, in the very best picture quality possible usually requires about 8 megabits per second, however very good to good quality HD picture quality can be achieved in much slower speeds even as low as between 3 and 4 megabits per second because the proprietary encoding technology of Sling Media is really that good.
Well I would advise you to consider the Quality and capabilities of your current router, and if it's older and cheaper, I would seriously recommend an upgrade to a far better more modern, more capable, more robust wi-fi performing router. It solved all of my problems.