How far can a planet with living beings get before we cannot see the living beings? Let there be a planet with Earth-like life. However, no trash or monuments or other visible structures, just organisms. How far can that planet be from Earth before our equipment just cannot observe the life anymore? Furthermore, let's say that our equipment is perfected and limited only by natural laws: how far can the planet be now?
The simple answer to this would be anywhere that is physically unreachable. If there happened to be Earth-like life out there in our solar system, we would likely be able to detect it, then land a probe and confirm. 

When it comes to planets outside our solar system, then it becomes difficult. If there was no trash orbiting (and thus no way to notice slight dips in light etc) then we would likely be limited to studying the atmosphere using the transit spectroscopy method (more on this HERE) where we can detect the gasses and chemical makeup of an exoplanets atmosphere. There may possibly be other things to look for in xray or infrared spectrums, such as localised hot-spots if there is intelligent life, although depending on distance and size of the planet, this could be difficult.

Overall, no matter what we detected, if it was to far away for us to physically be there, then we would only be theorising. We could say 'This planet is a great candidate for life' and things along those lines and present papers on findings. We would, however never be able to confirm it.