Only certain TVs from Roku themselves support Bluetooth headphones. You can only use the mobile app to get audio from Roku apps, not other devices connected to the TV, and the headphones are connected to the phone, not the TV.
As for a solution, you'd need some kind of device connected to the TV that has the capability of receiving audio from the TV and transmitting it to Bluetooth headphones connected to that device. You can find little devices that connect to the TV's headphone jack and then transmit via Bluetooth (a quick search of "tv bluetooth transmitter" on Amazon shows a few), but generally connecting something to the TV's headphone jack will disable the TV's speakers, so take that into account if you'd like the speakers to remain on. There's also devices that will connect to the TV's optical audio output assuming your TV has one.
I'm sure there's soundbars that transmit to connected Bluetooth devices, but be careful if you try to search for one. Many soundbars will receive audio via Bluetooth from a phone or computer, but they don't have the capability to transmit audio to a Bluetooth device, so the soundbar will advertise that it has Bluetooth connectivity, but it's not necessarily what you're looking for.
