For Those Using Cycles:
I think that raising divisions is definitely a good way to add detail to your fire. however, I would suggest keeping it below 200 if possible. Adding divisions will definitely slow down the render process.
Another way for you to add resolution to the fire is by enabling the Noise settings under the Domain's Physics settings. I would suggest you try this option before drastically raising your divisions. The Upres Factor I've heard is like multiplying the divisions amount by the number set in the field. If you enable this setting and set the Upres Factor to 2, it should help your fire look more realistic. I've been told that 2 is the highest you should bother to set it, as raising it higher does not add much more detail.
Hopefully by enabling the Noise property you will be able to raise the detail in your fire without drastically raising Render times.
Also, Be sure to bake the Fire settings and the Fire's Noise Settings.
For Those Using EVEE
If you are using EVEE, try this method before you start upping your fire's resolution divisions. Be advised that I am not using more than 64 resolution divisions in the following photos. My render samples are set at 64 for these pictures.
Here are the settings I am changing:
This picture has volumetric settings of 8px Tile Size and Samples of 86. As you can see (especially if you right-click and view image), it is pretty blocky.
This picture is set to Tile Size of 2px and Samples of 86. The fire has better detail.
This picture has a Tile Size of 2px and Samples of 256.
From what I've seen from these renders I have done is that lowering the Volumetrics Tile Size to 2x will greatly help improve the quality of your fire simulation - without you having to increase your fire resolution divisions to a huge amount. The volumetrics samples may add some detail to your fire, but 256 samples (which is the max amount) is probably overkill for your fire.
Try these settings before you start increasing your fire resolution divisions. Definitely lower the tile size to 2px, and mess around with the samples amount to see how it affects your fire. Overall, this should improve your fire without increasing your render time to such a huge amount as increasing divisions would.