Two important bottlenecks for a typical class AB amplifier are 1) the power supply PSU and 2) cooling facilities.
What you get in your sub 1K receiver is the PSU of a typical stereo integrated amp with 20,000-30,000uF caps, 300-500VA transformer. This is good enough for 50-80W continuous on 2 channels, 20Hz-20kHz @<1% THD. The heatsink in budget receivers is more or less same size as in a 50-80W stereo integrated amp.
However, they stack more output transistors to get more channels (5, 7...) on the same heatsink, powered by the same PSU as for stereo only. If you use it with only 2 channels, you are fine, no different than using a normal stereo integrated. The rest of the channels output devices are turned off and wont suck power from the PSU and generate heat. At this point there is no reason why AVR 7.1 should perform worse than 2.0 stereo amp. But for surround sound you may get clipping and overheating if pushing too hard simply because your 7 channel receiver amplification was built to a price of a 2 channel.
This is why when you look at receiver specs, look for continuous power on all channels, 20Hz-20kHz @<1% THD. Not peak power, not 1 channel driven, not 1kHz, not 10% THD.