There's no issue moving a Roku from one TV to another. I use my Stick on a different hotel TV every time I travel, including a trip next week.
I will say that your Roku 2 (there's five different versions of it) and your Roku 3 (two different versions) are both very old, and even if they are still a supported device (both 3s are, not all the 2s are) it won't be too much longer before they are unsupported. You might consider it time for an upgrade to a new model.
The remote for your 2 might work on the 3, but it's not certain. If your 2 is a 2 XS (model 3100) the remote will NOT work, as it's Bluetooth and it's the only Roku model that ever used Bluetooth for the remote.
If you're completely done with using your Roku 2, you need to first do a factory reset on the device, and then log into your Roku user account and remove it from your account. If your 2 and 3 are both linked to the same account, you won't have to do anything with the 3 when you move it to a different TV.
Dan
Roku Community Streaming Expert
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