How do I know if a tv signal amplifier** will also transmit IR signals
or maybe they are converted IR signals in the opposite direction?

**Like is used to go from a VCR to an distant analog TV, especially
if there were signal-weakening splitters for other tvs in between.

Rather than answering your question directly, I'm going to ask you to think
about what sort of hardware would be required to do that. You can then
figure it out for yourself, and you'll have learned something in the
process.