I have a Roku USB stick that was gifted to me around 2016.
The device works fine on my current HD TV but I am looking to purchase a new HDR 4K TV.
Will the device be slowed down on the new TV? Do I need to update the software on it? Should I purchase a new one?
The only Roku stick that can output a 4K image is the Streaming Stick+, model 3810, introduced in 2017 and still the top stick model in the line. All earlier sticks will still output 720p and 1080p HD images, and will still display on a 4K set. Most (all?) 4K sets can convert an HD input to near-4K quality for display (some are better at this than others) so you might be perfectly happy with your existing stick. I'd give it a go for a while before deciding to update, unless you also want to get some features in a later model.
Your old stick will not be slowed down on the new tv, it will run at the same speed you are seeing now. If your model is the 3500 (check via Settings > System > About), yeah, this is pretty slow and moving to a current model would give you a significant response speed boost on ANY tv. Plus some of the more recently created channel apps won't even run on the 3500.
Rokus should all check for updates daily if connected to a power source and automatically update to their current latest supported software version. You can force an update check manually via Settings > System > System update > Check now.
Roku Streaming Stick +, 3810X, Ser YH0059427035, wifi - Samsung UN55ES6100
Roku Streaming Stick, 3600X, Ser 5S56D8240827, wifi - JVC EM37T