I'd definitely pay in a recurring manner for new versions of JRemote on iOS in some way. There are many ways you can do this on the Apple AppStore (I can't really speak to what works well on Android).
1. Probably the best method is to release it as a limited free app with a subscription to unlock some set of the feature set. A simple target for this would be playback (the free-only version is truly just a "remote" unless you pay), though you could go for other things. This is the best, IMHO, because it is simple to understand, and on the Apple AppStore, you get 85% for subscription revenue after the first year (so you get more money too). Lots of "pro-quality" apps in the App Store have and are moving to this model.
2. A free app with an external required subscription or unlock on the "MC Side" of the equation (the mobile apps only work with the "Mobile Pack" add-on for MC, and otherwise you must use Panel). Ala Office 365, Adobe's apps, and Plex. This is more confusing for customers because they can't purchase the "fix" to the app from on the device, but... People are used to it for those other apps, and in this case, you have the benefit that you're already selling MC to customers each year. Even better, then you get to keep 100% of the funds (less your existing transaction fees). This will probably hurt adoption overall compared to option #1, but it probably fits with JRiver's overall current business model and "viewpoint". Plus it has the customer benefit of being able to keep old versions working without having to continue the subscription. And, there is competition doing basically the same thing in the same space. So...?
3. Do the same as #1 above, but with yearly in-app purchases to enable playback (and/or other features) with the new major version of desktop MC. This is less good than the previous suggestion because of the revenue thing and it'll be more confusing for customers. But, the upside for the customer is that you can stop the upgrade train if you want and stick with what you've got (like MC on the desktop works now).
4. Release new versions each year, tied to the major version of MC that they work with. I think this is a bad idea, and will make customers grumpy, but it's an option.
If I could have offline playback and improvements to avoid having to use a separate app to stream video to my devices, I'd definitely be in for any of these options. As it is, I'm strongly considering going to Plex. Over the past 18 months or so, a LOT of our media consumption has moved to mobile devices. This is, honestly, my biggest concern about my JRiver-based setup right now. I still use the HTPC, but for basically everything else (which is a lot) I'm using AirVideo HD (which is unsupported and annoying) and a smattering of streaming apps (which I don't really like, but they work).