The converter is, of course, a non-standard hookup. Having said that, I don't know of any reason it would cause the Roku Streaming Stick+ (that's what your device is) to work extra hard.
An alternate option would be to get a Roku Express to use with the adapter and that TV. You should be able to download the YouTube TV app despite the contract dispute. It may take some work, but I'd suggest that route if you don't want to replace your TV.
Of course, if you want a better Roku, get an Ultra. It costs more. The Express is a lesser device, but should work for what you're after. It's ⅓ the price of an Ultra and ½ the price of a Stick+.
A new TV is still a better idea, but I am offering those other options if you want to consider them.
DBDukes
Roku Community Streaming Expert
Note: I am not a Roku employee.
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Dear All:
My Roku box measures 3 1/4 x 1 1/2 and sits flat on wood,
which warms up and sometimes the overheating message
pops up for a few seconds.
There is no heat source around and the ambient temperature
is normal (family room).
I placed a very flat piece of metal, a little larger than the box,
under the box. Since then the heat transfer into the metal and
the wood is very good, and the overheat message no longer
pops up.
Do not put the metal on top of the box, it can screw your WiFi
reception (mine did so).
Best wishes
Henning